mathlib documentation

group_theory.subgroup.basic

Subgroups #

This file defines multiplicative and additive subgroups as an extension of submonoids, in a bundled form (unbundled subgroups are in deprecated/subgroups.lean).

We prove subgroups of a group form a complete lattice, and results about images and preimages of subgroups under group homomorphisms. The bundled subgroups use bundled monoid homomorphisms.

There are also theorems about the subgroups generated by an element or a subset of a group, defined both inductively and as the infimum of the set of subgroups containing a given element/subset.

Special thanks goes to Amelia Livingston and Yury Kudryashov for their help and inspiration.

Main definitions #

Notation used here:

Definitions in the file:

Implementation notes #

Subgroup inclusion is denoted rather than , although is defined as membership of a subgroup's underlying set.

Tags #

subgroup, subgroups

def subgroup.to_submonoid {G : Type u_3} [group G] (self : subgroup G) :

Reinterpret a subgroup as a submonoid.

structure subgroup (G : Type u_3) [group G] :
Type u_3

A subgroup of a group G is a subset containing 1, closed under multiplication and closed under multiplicative inverse.

structure add_subgroup (G : Type u_3) [add_group G] :
Type u_3

An additive subgroup of an additive group G is a subset containing 0, closed under addition and additive inverse.

def add_subgroup.to_add_submonoid {G : Type u_3} [add_group G] (self : add_subgroup G) :

Reinterpret an add_subgroup as an add_submonoid.

@[protected, instance]
def subgroup.set_like {G : Type u_1} [group G] :
Equations
@[protected, instance]
def add_subgroup.set_like {G : Type u_1} [add_group G] :
Equations
@[simp]
theorem subgroup.mem_carrier {G : Type u_1} [group G] {s : subgroup G} {x : G} :
x s.carrier x s
@[simp]
theorem add_subgroup.mem_carrier {G : Type u_1} [add_group G] {s : add_subgroup G} {x : G} :
x s.carrier x s
@[simp]
theorem add_subgroup.mem_mk {G : Type u_1} [add_group G] {s : set G} {x : G} (h_one : 0 s) (h_mul : ∀ {a b : G}, a sb sa + b s) (h_inv : ∀ {x : G}, x s-x s) :
x {carrier := s, zero_mem' := h_one, add_mem' := h_mul, neg_mem' := h_inv} x s
@[simp]
theorem subgroup.mem_mk {G : Type u_1} [group G] {s : set G} {x : G} (h_one : 1 s) (h_mul : ∀ {a b : G}, a sb sa * b s) (h_inv : ∀ {x : G}, x sx⁻¹ s) :
x {carrier := s, one_mem' := h_one, mul_mem' := h_mul, inv_mem' := h_inv} x s
@[simp]
theorem subgroup.coe_set_mk {G : Type u_1} [group G] {s : set G} (h_one : 1 s) (h_mul : ∀ {a b : G}, a sb sa * b s) (h_inv : ∀ {x : G}, x sx⁻¹ s) :
{carrier := s, one_mem' := h_one, mul_mem' := h_mul, inv_mem' := h_inv} = s
@[simp]
theorem add_subgroup.coe_set_mk {G : Type u_1} [add_group G] {s : set G} (h_one : 0 s) (h_mul : ∀ {a b : G}, a sb sa + b s) (h_inv : ∀ {x : G}, x s-x s) :
{carrier := s, zero_mem' := h_one, add_mem' := h_mul, neg_mem' := h_inv} = s
@[simp]
theorem add_subgroup.mk_le_mk {G : Type u_1} [add_group G] {s t : set G} (h_one : 0 s) (h_mul : ∀ {a b : G}, a sb sa + b s) (h_inv : ∀ {x : G}, x s-x s) (h_one' : 0 t) (h_mul' : ∀ {a b : G}, a tb ta + b t) (h_inv' : ∀ {x : G}, x t-x t) :
{carrier := s, zero_mem' := h_one, add_mem' := h_mul, neg_mem' := h_inv} {carrier := t, zero_mem' := h_one', add_mem' := h_mul', neg_mem' := h_inv'} s t
@[simp]
theorem subgroup.mk_le_mk {G : Type u_1} [group G] {s t : set G} (h_one : 1 s) (h_mul : ∀ {a b : G}, a sb sa * b s) (h_inv : ∀ {x : G}, x sx⁻¹ s) (h_one' : 1 t) (h_mul' : ∀ {a b : G}, a tb ta * b t) (h_inv' : ∀ {x : G}, x tx⁻¹ t) :
{carrier := s, one_mem' := h_one, mul_mem' := h_mul, inv_mem' := h_inv} {carrier := t, one_mem' := h_one', mul_mem' := h_mul', inv_mem' := h_inv'} s t
def subgroup.simps.coe {G : Type u_1} [group G] (S : subgroup G) :
set G

See Note [custom simps projection]

Equations
@[simp]
@[simp]
theorem subgroup.coe_to_submonoid {G : Type u_1} [group G] (K : subgroup G) :
@[simp]
theorem subgroup.mem_to_submonoid {G : Type u_1} [group G] (K : subgroup G) (x : G) :
@[simp]
theorem add_subgroup.mem_to_add_submonoid {G : Type u_1} [add_group G] (K : add_subgroup G) (x : G) :
@[protected, instance]
def subgroup.fintype {G : Type u_1} [group G] (K : subgroup G) [d : decidable_pred (λ (_x : G), _x K)] [fintype G] :
Equations
@[protected, instance]
def add_subgroup.fintype {G : Type u_1} [add_group G] (K : add_subgroup G) [d : decidable_pred (λ (_x : G), _x K)] [fintype G] :
Equations
@[simp]
@[simp]
theorem subgroup.to_submonoid_eq {G : Type u_1} [group G] {p q : subgroup G} :
@[simp]
theorem subgroup.to_submonoid_le {G : Type u_1} [group G] {p q : subgroup G} :

Conversion to/from additive/multiplicative #

Supgroups of a group G are isomorphic to additive subgroups of additive G.

Equations

Additive subgroup of an additive group additive G are isomorphic to subgroup of G.

Additive supgroups of an additive group A are isomorphic to subgroups of multiplicative A.

Equations

Subgroups of an additive group multiplicative A are isomorphic to additive subgroups of A.

@[protected]
def subgroup.copy {G : Type u_1} [group G] (K : subgroup G) (s : set G) (hs : s = K) :

Copy of a subgroup with a new carrier equal to the old one. Useful to fix definitional equalities.

Equations
@[protected]
def add_subgroup.copy {G : Type u_1} [add_group G] (K : add_subgroup G) (s : set G) (hs : s = K) :

Copy of an additive subgroup with a new carrier equal to the old one. Useful to fix definitional equalities

Equations
@[simp]
theorem add_subgroup.coe_copy {G : Type u_1} [add_group G] (K : add_subgroup G) (s : set G) (hs : s = K) :
(K.copy s hs) = s
@[simp]
theorem subgroup.coe_copy {G : Type u_1} [group G] (K : subgroup G) (s : set G) (hs : s = K) :
(K.copy s hs) = s
theorem add_subgroup.copy_eq {G : Type u_1} [add_group G] (K : add_subgroup G) (s : set G) (hs : s = K) :
K.copy s hs = K
theorem subgroup.copy_eq {G : Type u_1} [group G] (K : subgroup G) (s : set G) (hs : s = K) :
K.copy s hs = K
@[ext]
theorem subgroup.ext {G : Type u_1} [group G] {H K : subgroup G} (h : ∀ (x : G), x H x K) :
H = K

Two subgroups are equal if they have the same elements.

theorem add_subgroup.ext {G : Type u_1} [add_group G] {H K : add_subgroup G} (h : ∀ (x : G), x H x K) :
H = K

Two add_subgroups are equal if they have the same elements.

theorem subgroup.one_mem {G : Type u_1} [group G] (H : subgroup G) :
1 H

A subgroup contains the group's 1.

theorem add_subgroup.zero_mem {G : Type u_1} [add_group G] (H : add_subgroup G) :
0 H

An add_subgroup contains the group's 0.

theorem add_subgroup.add_mem {G : Type u_1} [add_group G] (H : add_subgroup G) {x y : G} :
x Hy Hx + y H

An add_subgroup is closed under addition.

theorem subgroup.mul_mem {G : Type u_1} [group G] (H : subgroup G) {x y : G} :
x Hy Hx * y H

A subgroup is closed under multiplication.

theorem subgroup.inv_mem {G : Type u_1} [group G] (H : subgroup G) {x : G} :
x Hx⁻¹ H

A subgroup is closed under inverse.

theorem add_subgroup.neg_mem {G : Type u_1} [add_group G] (H : add_subgroup G) {x : G} :
x H-x H

An add_subgroup is closed under inverse.

theorem subgroup.div_mem {G : Type u_1} [group G] (H : subgroup G) {x y : G} (hx : x H) (hy : y H) :
x / y H

A subgroup is closed under division.

theorem add_subgroup.sub_mem {G : Type u_1} [add_group G] (H : add_subgroup G) {x y : G} (hx : x H) (hy : y H) :
x - y H

An add_subgroup is closed under subtraction.

@[simp]
theorem add_subgroup.neg_mem_iff {G : Type u_1} [add_group G] (H : add_subgroup G) {x : G} :
-x H x H
@[simp]
theorem subgroup.inv_mem_iff {G : Type u_1} [group G] (H : subgroup G) {x : G} :
x⁻¹ H x H
theorem add_subgroup.sub_mem_comm_iff {G : Type u_1} [add_group G] (H : add_subgroup G) {a b : G} :
a - b H b - a H
theorem subgroup.div_mem_comm_iff {G : Type u_1} [group G] (H : subgroup G) {a b : G} :
a / b H b / a H
@[simp]
theorem add_subgroup.neg_coe_set {G : Type u_1} [add_group G] (H : add_subgroup G) :
@[simp]
theorem subgroup.inv_coe_set {G : Type u_1} [group G] (H : subgroup G) :
@[simp]
theorem subgroup.exists_inv_mem_iff_exists_mem {G : Type u_1} [group G] (K : subgroup G) {P : G → Prop} :
(∃ (x : G), x K P x⁻¹) ∃ (x : G) (H : x K), P x
@[simp]
theorem add_subgroup.exists_neg_mem_iff_exists_mem {G : Type u_1} [add_group G] (K : add_subgroup G) {P : G → Prop} :
(∃ (x : G), x K P (-x)) ∃ (x : G) (H : x K), P x
theorem add_subgroup.add_mem_cancel_right {G : Type u_1} [add_group G] (H : add_subgroup G) {x y : G} (h : x H) :
y + x H y H
theorem subgroup.mul_mem_cancel_right {G : Type u_1} [group G] (H : subgroup G) {x y : G} (h : x H) :
y * x H y H
theorem subgroup.mul_mem_cancel_left {G : Type u_1} [group G] (H : subgroup G) {x y : G} (h : x H) :
x * y H y H
theorem add_subgroup.add_mem_cancel_left {G : Type u_1} [add_group G] (H : add_subgroup G) {x y : G} (h : x H) :
x + y H y H
theorem subgroup.list_prod_mem {G : Type u_1} [group G] (K : subgroup G) {l : list G} :
(∀ (x : G), x lx K)l.prod K

Product of a list of elements in a subgroup is in the subgroup.

theorem add_subgroup.list_sum_mem {G : Type u_1} [add_group G] (K : add_subgroup G) {l : list G} :
(∀ (x : G), x lx K)l.sum K

Sum of a list of elements in an add_subgroup is in the add_subgroup.

theorem add_subgroup.multiset_sum_mem {G : Type u_1} [add_comm_group G] (K : add_subgroup G) (g : multiset G) :
(∀ (a : G), a ga K)g.sum K

Sum of a multiset of elements in an add_subgroup of an add_comm_group is in the add_subgroup.

theorem subgroup.multiset_prod_mem {G : Type u_1} [comm_group G] (K : subgroup G) (g : multiset G) :
(∀ (a : G), a ga K)g.prod K

Product of a multiset of elements in a subgroup of a comm_group is in the subgroup.

theorem add_subgroup.sum_mem {G : Type u_1} [add_comm_group G] (K : add_subgroup G) {ι : Type u_2} {t : finset ι} {f : ι → G} (h : ∀ (c : ι), c tf c K) :
∑ (c : ι) in t, f c K

Sum of elements in an add_subgroup of an add_comm_group indexed by a finset is in the add_subgroup.

theorem subgroup.prod_mem {G : Type u_1} [comm_group G] (K : subgroup G) {ι : Type u_2} {t : finset ι} {f : ι → G} (h : ∀ (c : ι), c tf c K) :
∏ (c : ι) in t, f c K

Product of elements of a subgroup of a comm_group indexed by a finset is in the subgroup.

theorem subgroup.pow_mem {G : Type u_1} [group G] (K : subgroup G) {x : G} (hx : x K) (n : ) :
x ^ n K
theorem add_subgroup.nsmul_mem {G : Type u_1} [add_group G] (K : add_subgroup G) {x : G} (hx : x K) (n : ) :
n x K
theorem subgroup.zpow_mem {G : Type u_1} [group G] (K : subgroup G) {x : G} (hx : x K) (n : ) :
x ^ n K
theorem add_subgroup.zsmul_mem {G : Type u_1} [add_group G] (K : add_subgroup G) {x : G} (hx : x K) (n : ) :
n x K
def subgroup.of_div {G : Type u_1} [group G] (s : set G) (hsn : s.nonempty) (hs : ∀ (x y : G), x sy sx * y⁻¹ s) :

Construct a subgroup from a nonempty set that is closed under division.

Equations
def add_subgroup.of_sub {G : Type u_1} [add_group G] (s : set G) (hsn : s.nonempty) (hs : ∀ (x y : G), x sy sx + -y s) :

Construct a subgroup from a nonempty set that is closed under subtraction

Equations
@[protected, instance]
def subgroup.has_mul {G : Type u_1} [group G] (H : subgroup G) :

A subgroup of a group inherits a multiplication.

Equations
@[protected, instance]
def add_subgroup.has_add {G : Type u_1} [add_group G] (H : add_subgroup G) :

An add_subgroup of an add_group inherits an addition.

Equations
@[protected, instance]
def subgroup.has_one {G : Type u_1} [group G] (H : subgroup G) :

A subgroup of a group inherits a 1.

Equations
@[protected, instance]
def add_subgroup.has_zero {G : Type u_1} [add_group G] (H : add_subgroup G) :

An add_subgroup of an add_group inherits a zero.

Equations
@[protected, instance]
def add_subgroup.has_neg {G : Type u_1} [add_group G] (H : add_subgroup G) :

A add_subgroup of a add_group inherits an inverse.

Equations
@[protected, instance]
def subgroup.has_inv {G : Type u_1} [group G] (H : subgroup G) :

A subgroup of a group inherits an inverse.

Equations
@[protected, instance]
def subgroup.has_div {G : Type u_1} [group G] (H : subgroup G) :

A subgroup of a group inherits a division

Equations
@[protected, instance]
def add_subgroup.has_sub {G : Type u_1} [add_group G] (H : add_subgroup G) :

An add_subgroup of an add_group inherits a subtraction.

Equations
@[simp, norm_cast]
theorem add_subgroup.coe_add {G : Type u_1} [add_group G] (H : add_subgroup G) (x y : H) :
(x + y) = x + y
@[simp, norm_cast]
theorem subgroup.coe_mul {G : Type u_1} [group G] (H : subgroup G) (x y : H) :
x * y = (x) * y
@[simp, norm_cast]
theorem add_subgroup.coe_zero {G : Type u_1} [add_group G] (H : add_subgroup G) :
0 = 0
@[simp, norm_cast]
theorem subgroup.coe_one {G : Type u_1} [group G] (H : subgroup G) :
1 = 1
@[simp, norm_cast]
theorem add_subgroup.coe_neg {G : Type u_1} [add_group G] (H : add_subgroup G) (x : H) :
@[simp, norm_cast]
theorem subgroup.coe_inv {G : Type u_1} [group G] (H : subgroup G) (x : H) :
@[simp, norm_cast]
theorem subgroup.coe_div {G : Type u_1} [group G] (H : subgroup G) (x y : H) :
(x / y) = x / y
@[simp, norm_cast]
theorem add_subgroup.coe_sub {G : Type u_1} [add_group G] (H : add_subgroup G) (x y : H) :
(x - y) = x - y
@[simp, norm_cast]
theorem add_subgroup.coe_mk {G : Type u_1} [add_group G] (H : add_subgroup G) (x : G) (hx : x H) :
x, hx⟩ = x
@[simp, norm_cast]
theorem subgroup.coe_mk {G : Type u_1} [group G] (H : subgroup G) (x : G) (hx : x H) :
x, hx⟩ = x
@[protected, instance]
def subgroup.to_group {G : Type u_1} [group G] (H : subgroup G) :

A subgroup of a group inherits a group structure.

Equations
@[protected, instance]
def add_subgroup.to_add_group {G : Type u_1} [add_group G] (H : add_subgroup G) :

An add_subgroup of an add_group inherits an add_group structure.

Equations
@[protected, instance]
def subgroup.to_comm_group {G : Type u_1} [comm_group G] (H : subgroup G) :

A subgroup of a comm_group is a comm_group.

Equations
@[protected, instance]

An add_subgroup of an add_ordered_comm_group is an add_ordered_comm_group.

Equations
def subgroup.subtype {G : Type u_1} [group G] (H : subgroup G) :

The natural group hom from a subgroup of group G to G.

Equations
def add_subgroup.subtype {G : Type u_1} [add_group G] (H : add_subgroup G) :

The natural group hom from an add_subgroup of add_group G to G.

Equations
@[simp]
theorem subgroup.coe_subtype {G : Type u_1} [group G] (H : subgroup G) :
@[simp]
theorem add_subgroup.coe_subtype {G : Type u_1} [add_group G] (H : add_subgroup G) :
@[simp, norm_cast]
theorem add_subgroup.coe_smul {G : Type u_1} [add_group G] (H : add_subgroup G) (x : H) (n : ) :
(n x) = n x
@[simp, norm_cast]
theorem subgroup.coe_pow {G : Type u_1} [group G] (H : subgroup G) (x : H) (n : ) :
(x ^ n) = x ^ n
@[simp, norm_cast]
theorem add_subgroup.coe_zsmul {G : Type u_1} [add_group G] (H : add_subgroup G) (x : H) (n : ) :
(n x) = n x
@[simp, norm_cast]
theorem subgroup.coe_zpow {G : Type u_1} [group G] (H : subgroup G) (x : H) (n : ) :
(x ^ n) = x ^ n
@[simp, norm_cast]
theorem add_subgroup.coe_list_sum {G : Type u_1} [add_group G] (H : add_subgroup G) (l : list H) :
@[simp, norm_cast]
theorem subgroup.coe_list_prod {G : Type u_1} [group G] (H : subgroup G) (l : list H) :
@[simp, norm_cast]
theorem subgroup.coe_multiset_prod {G : Type u_1} [comm_group G] (H : subgroup G) (m : multiset H) :
@[simp, norm_cast]
theorem add_subgroup.coe_multiset_sum {G : Type u_1} [add_comm_group G] (H : add_subgroup G) (m : multiset H) :
@[simp, norm_cast]
theorem add_subgroup.coe_finset_sum {ι : Type u_1} {G : Type u_2} [add_comm_group G] (H : add_subgroup G) (f : ι → H) (s : finset ι) :
∑ (i : ι) in s, f i = ∑ (i : ι) in s, (f i)
@[simp, norm_cast]
theorem subgroup.coe_finset_prod {ι : Type u_1} {G : Type u_2} [comm_group G] (H : subgroup G) (f : ι → H) (s : finset ι) :
∏ (i : ι) in s, f i = ∏ (i : ι) in s, (f i)
def add_subgroup.inclusion {G : Type u_1} [add_group G] {H K : add_subgroup G} (h : H K) :

The inclusion homomorphism from a additive subgroup H contained in K to K.

Equations
def subgroup.inclusion {G : Type u_1} [group G] {H K : subgroup G} (h : H K) :

The inclusion homomorphism from a subgroup H contained in K to K.

Equations
@[simp]
theorem subgroup.coe_inclusion {G : Type u_1} [group G] {H K : subgroup G} {h : H K} (a : H) :
@[simp]
theorem add_subgroup.coe_inclusion {G : Type u_1} [add_group G] {H K : add_subgroup G} {h : H K} (a : H) :
@[simp]
theorem add_subgroup.subtype_comp_inclusion {G : Type u_1} [add_group G] {H K : add_subgroup G} (hH : H K) :
@[simp]
theorem subgroup.subtype_comp_inclusion {G : Type u_1} [group G] {H K : subgroup G} (hH : H K) :
@[protected, instance]
def add_subgroup.has_top {G : Type u_1} [add_group G] :

The add_subgroup G of the add_group G.

Equations
@[protected, instance]
def subgroup.has_top {G : Type u_1} [group G] :

The subgroup G of the group G.

Equations
@[protected, instance]
def add_subgroup.has_bot {G : Type u_1} [add_group G] :

The trivial add_subgroup {0} of an add_group G.

Equations
@[protected, instance]
def subgroup.has_bot {G : Type u_1} [group G] :

The trivial subgroup {1} of an group G.

Equations
@[protected, instance]
def add_subgroup.inhabited {G : Type u_1} [add_group G] :
Equations
@[protected, instance]
def subgroup.inhabited {G : Type u_1} [group G] :
Equations
@[simp]
theorem subgroup.mem_bot {G : Type u_1} [group G] {x : G} :
x x = 1
@[simp]
theorem add_subgroup.mem_bot {G : Type u_1} [add_group G] {x : G} :
x x = 0
@[simp]
theorem add_subgroup.mem_top {G : Type u_1} [add_group G] (x : G) :
@[simp]
theorem subgroup.mem_top {G : Type u_1} [group G] (x : G) :
@[simp]
theorem subgroup.coe_top {G : Type u_1} [group G] :
@[simp]
theorem add_subgroup.coe_top {G : Type u_1} [add_group G] :
@[simp]
theorem add_subgroup.coe_bot {G : Type u_1} [add_group G] :
= {0}
@[simp]
theorem subgroup.coe_bot {G : Type u_1} [group G] :
= {1}
@[protected, instance]
Equations
@[protected, instance]
def subgroup.has_bot.bot.unique {G : Type u_1} [group G] :
Equations
theorem subgroup.eq_bot_iff_forall {G : Type u_1} [group G] (H : subgroup G) :
H = ∀ (x : G), x Hx = 1
theorem add_subgroup.eq_bot_iff_forall {G : Type u_1} [add_group G] (H : add_subgroup G) :
H = ∀ (x : G), x Hx = 0
theorem subgroup.eq_bot_of_subsingleton {G : Type u_1} [group G] (H : subgroup G) [subsingleton H] :
H =
theorem subgroup.coe_eq_univ {G : Type u_1} [group G] {H : subgroup G} :
theorem add_subgroup.coe_eq_univ {G : Type u_1} [add_group G] {H : add_subgroup G} :
theorem add_subgroup.coe_eq_singleton {G : Type u_1} [add_group G] {H : add_subgroup G} :
(∃ (g : G), H = {g}) H =
theorem subgroup.coe_eq_singleton {G : Type u_1} [group G] {H : subgroup G} :
(∃ (g : G), H = {g}) H =
@[protected, instance]
def subgroup.fintype_bot {G : Type u_1} [group G] :
Equations
@[protected, instance]
def add_subgroup.fintype_bot {G : Type u_1} [add_group G] :
Equations
@[simp]
theorem subgroup.card_bot {G : Type u_1} [group G] {_x : fintype } :
@[simp]
theorem add_subgroup.card_bot {G : Type u_1} [add_group G] {_x : fintype } :
theorem subgroup.eq_top_of_card_eq {G : Type u_1} [group G] (H : subgroup G) [fintype H] [fintype G] (h : fintype.card H = fintype.card G) :
H =
theorem subgroup.eq_top_of_le_card {G : Type u_1} [group G] (H : subgroup G) [fintype H] [fintype G] (h : fintype.card G fintype.card H) :
H =
theorem subgroup.eq_bot_of_card_le {G : Type u_1} [group G] (H : subgroup G) [fintype H] (h : fintype.card H 1) :
H =
theorem add_subgroup.eq_bot_of_card_le {G : Type u_1} [add_group G] (H : add_subgroup G) [fintype H] (h : fintype.card H 1) :
H =
theorem add_subgroup.eq_bot_of_card_eq {G : Type u_1} [add_group G] (H : add_subgroup G) [fintype H] (h : fintype.card H = 1) :
H =
theorem subgroup.eq_bot_of_card_eq {G : Type u_1} [group G] (H : subgroup G) [fintype H] (h : fintype.card H = 1) :
H =
theorem add_subgroup.nontrivial_iff_exists_ne_zero {G : Type u_1} [add_group G] (H : add_subgroup G) :
nontrivial H ∃ (x : G) (H : x H), x 0
theorem subgroup.nontrivial_iff_exists_ne_one {G : Type u_1} [group G] (H : subgroup G) :
nontrivial H ∃ (x : G) (H : x H), x 1
theorem subgroup.bot_or_nontrivial {G : Type u_1} [group G] (H : subgroup G) :

A subgroup is either the trivial subgroup or nontrivial.

theorem subgroup.bot_or_exists_ne_one {G : Type u_1} [group G] (H : subgroup G) :
H = ∃ (x : G) (H : x H), x 1

A subgroup is either the trivial subgroup or contains a nonzero element.

theorem add_subgroup.bot_or_exists_ne_zero {G : Type u_1} [add_group G] (H : add_subgroup G) :
H = ∃ (x : G) (H : x H), x 0
theorem subgroup.card_le_one_iff_eq_bot {G : Type u_1} [group G] (H : subgroup G) [fintype H] :
theorem subgroup.one_lt_card_iff_ne_bot {G : Type u_1} [group G] (H : subgroup G) [fintype H] :
@[protected, instance]
def subgroup.has_inf {G : Type u_1} [group G] :

The inf of two subgroups is their intersection.

Equations
@[protected, instance]
def add_subgroup.has_inf {G : Type u_1} [add_group G] :

The inf of two add_subgroupss is their intersection.

Equations
@[simp]
theorem subgroup.coe_inf {G : Type u_1} [group G] (p p' : subgroup G) :
(p p') = p p'
@[simp]
theorem add_subgroup.coe_inf {G : Type u_1} [add_group G] (p p' : add_subgroup G) :
(p p') = p p'
@[simp]
theorem add_subgroup.mem_inf {G : Type u_1} [add_group G] {p p' : add_subgroup G} {x : G} :
x p p' x p x p'
@[simp]
theorem subgroup.mem_inf {G : Type u_1} [group G] {p p' : subgroup G} {x : G} :
x p p' x p x p'
@[protected, instance]
def add_subgroup.has_Inf {G : Type u_1} [add_group G] :
Equations
@[protected, instance]
def subgroup.has_Inf {G : Type u_1} [group G] :
Equations
@[simp, norm_cast]
theorem subgroup.coe_Inf {G : Type u_1} [group G] (H : set (subgroup G)) :
(Inf H) = ⋂ (s : subgroup G) (H : s H), s
@[simp, norm_cast]
theorem add_subgroup.coe_Inf {G : Type u_1} [add_group G] (H : set (add_subgroup G)) :
(Inf H) = ⋂ (s : add_subgroup G) (H : s H), s
@[simp]
theorem add_subgroup.mem_Inf {G : Type u_1} [add_group G] {S : set (add_subgroup G)} {x : G} :
x Inf S ∀ (p : add_subgroup G), p Sx p
@[simp]
theorem subgroup.mem_Inf {G : Type u_1} [group G] {S : set (subgroup G)} {x : G} :
x Inf S ∀ (p : subgroup G), p Sx p
theorem subgroup.mem_infi {G : Type u_1} [group G] {ι : Sort u_2} {S : ι → subgroup G} {x : G} :
(x ⨅ (i : ι), S i) ∀ (i : ι), x S i
theorem add_subgroup.mem_infi {G : Type u_1} [add_group G] {ι : Sort u_2} {S : ι → add_subgroup G} {x : G} :
(x ⨅ (i : ι), S i) ∀ (i : ι), x S i
@[simp, norm_cast]
theorem subgroup.coe_infi {G : Type u_1} [group G] {ι : Sort u_2} {S : ι → subgroup G} :
(⨅ (i : ι), S i) = ⋂ (i : ι), (S i)
@[simp, norm_cast]
theorem add_subgroup.coe_infi {G : Type u_1} [add_group G] {ι : Sort u_2} {S : ι → add_subgroup G} :
(⨅ (i : ι), S i) = ⋂ (i : ι), (S i)
@[protected, instance]

Subgroups of a group form a complete lattice.

Equations
@[protected, instance]

The add_subgroups of an add_group form a complete lattice.

Equations
theorem subgroup.mem_sup_left {G : Type u_1} [group G] {S T : subgroup G} {x : G} :
x Sx S T
theorem add_subgroup.mem_sup_left {G : Type u_1} [add_group G] {S T : add_subgroup G} {x : G} :
x Sx S T
theorem subgroup.mem_sup_right {G : Type u_1} [group G] {S T : subgroup G} {x : G} :
x Tx S T
theorem add_subgroup.mem_sup_right {G : Type u_1} [add_group G] {S T : add_subgroup G} {x : G} :
x Tx S T
theorem subgroup.mul_mem_sup {G : Type u_1} [group G] {S T : subgroup G} {x y : G} (hx : x S) (hy : y T) :
x * y S T
theorem add_subgroup.add_mem_sup {G : Type u_1} [add_group G] {S T : add_subgroup G} {x y : G} (hx : x S) (hy : y T) :
x + y S T
theorem subgroup.mem_supr_of_mem {G : Type u_1} [group G] {ι : Type u_2} {S : ι → subgroup G} (i : ι) {x : G} :
x S ix supr S
theorem add_subgroup.mem_supr_of_mem {G : Type u_1} [add_group G] {ι : Type u_2} {S : ι → add_subgroup G} (i : ι) {x : G} :
x S ix supr S
theorem subgroup.mem_Sup_of_mem {G : Type u_1} [group G] {S : set (subgroup G)} {s : subgroup G} (hs : s S) {x : G} :
x sx Sup S
theorem add_subgroup.mem_Sup_of_mem {G : Type u_1} [add_group G] {S : set (add_subgroup G)} {s : add_subgroup G} (hs : s S) {x : G} :
x sx Sup S
@[simp]
@[simp]
theorem subgroup.nontrivial_iff {G : Type u_1} [group G] :
@[simp]
@[protected, instance]
def add_subgroup.unique {G : Type u_1} [add_group G] [subsingleton G] :
Equations
@[protected, instance]
def subgroup.unique {G : Type u_1} [group G] [subsingleton G] :
Equations
@[protected, instance]
def subgroup.nontrivial {G : Type u_1} [group G] [nontrivial G] :
@[protected, instance]
theorem add_subgroup.eq_top_iff' {G : Type u_1} [add_group G] (H : add_subgroup G) :
H = ∀ (x : G), x H
theorem subgroup.eq_top_iff' {G : Type u_1} [group G] (H : subgroup G) :
H = ∀ (x : G), x H
def add_subgroup.closure {G : Type u_1} [add_group G] (k : set G) :

The add_subgroup generated by a set

Equations
def subgroup.closure {G : Type u_1} [group G] (k : set G) :

The subgroup generated by a set.

Equations
theorem subgroup.mem_closure {G : Type u_1} [group G] {k : set G} {x : G} :
x subgroup.closure k ∀ (K : subgroup G), k Kx K
theorem add_subgroup.mem_closure {G : Type u_1} [add_group G] {k : set G} {x : G} :
x add_subgroup.closure k ∀ (K : add_subgroup G), k Kx K
@[simp]
theorem subgroup.subset_closure {G : Type u_1} [group G] {k : set G} :

The subgroup generated by a set includes the set.

@[simp]
theorem add_subgroup.subset_closure {G : Type u_1} [add_group G] {k : set G} :

The add_subgroup generated by a set includes the set.

theorem add_subgroup.not_mem_of_not_mem_closure {G : Type u_1} [add_group G] {k : set G} {P : G} (hP : P add_subgroup.closure k) :
P k
theorem subgroup.not_mem_of_not_mem_closure {G : Type u_1} [group G] {k : set G} {P : G} (hP : P subgroup.closure k) :
P k
@[simp]
theorem subgroup.closure_le {G : Type u_1} [group G] (K : subgroup G) {k : set G} :

A subgroup K includes closure k if and only if it includes k.

@[simp]
theorem add_subgroup.closure_le {G : Type u_1} [add_group G] (K : add_subgroup G) {k : set G} :

An additive subgroup K includes closure k if and only if it includes k

theorem subgroup.closure_eq_of_le {G : Type u_1} [group G] (K : subgroup G) {k : set G} (h₁ : k K) (h₂ : K subgroup.closure k) :
theorem add_subgroup.closure_eq_of_le {G : Type u_1} [add_group G] (K : add_subgroup G) {k : set G} (h₁ : k K) (h₂ : K add_subgroup.closure k) :
theorem subgroup.closure_induction {G : Type u_1} [group G] {k : set G} {p : G → Prop} {x : G} (h : x subgroup.closure k) (Hk : ∀ (x : G), x kp x) (H1 : p 1) (Hmul : ∀ (x y : G), p xp yp (x * y)) (Hinv : ∀ (x : G), p xp x⁻¹) :
p x

An induction principle for closure membership. If p holds for 1 and all elements of k, and is preserved under multiplication and inverse, then p holds for all elements of the closure of k.

theorem add_subgroup.closure_induction {G : Type u_1} [add_group G] {k : set G} {p : G → Prop} {x : G} (h : x add_subgroup.closure k) (Hk : ∀ (x : G), x kp x) (H1 : p 0) (Hmul : ∀ (x y : G), p xp yp (x + y)) (Hinv : ∀ (x : G), p xp (-x)) :
p x

An induction principle for additive closure membership. If p holds for 0 and all elements of k, and is preserved under addition and inverses, then p holds for all elements of the additive closure of k.

theorem subgroup.closure_induction' {G : Type u_1} [group G] (k : set G) {p : (subgroup.closure k) → Prop} (Hk : ∀ (x : G) (h : x k), p x, _⟩) (H1 : p 1) (Hmul : ∀ (x y : (subgroup.closure k)), p xp yp (x * y)) (Hinv : ∀ (x : (subgroup.closure k)), p xp x⁻¹) (x : (subgroup.closure k)) :
p x

An induction principle on elements of the subtype subgroup.closure. If p holds for 1 and all elements of k, and is preserved under multiplication and inverse, then p holds for all elements x : closure k.

The difference with subgroup.closure_induction is that this acts on the subtype.

theorem add_subgroup.closure_induction' {G : Type u_1} [add_group G] (k : set G) {p : (add_subgroup.closure k) → Prop} (Hk : ∀ (x : G) (h : x k), p x, _⟩) (H1 : p 0) (Hmul : ∀ (x y : (add_subgroup.closure k)), p xp yp (x + y)) (Hinv : ∀ (x : (add_subgroup.closure k)), p xp (-x)) (x : (add_subgroup.closure k)) :
p x

An induction principle on elements of the subtype add_subgroup.closure. If p holds for 0 and all elements of k, and is preserved under addition and negation, then p holds for all elements x : closure k.

The difference with add_subgroup.closure_induction is that this acts on the subtype.

@[simp]
theorem subgroup.closure_closure_coe_preimage {G : Type u_1} [group G] {k : set G} :
@[protected]

closure forms a Galois insertion with the coercion to set.

Equations
@[protected]

closure forms a Galois insertion with the coercion to set.

Equations
theorem subgroup.closure_mono {G : Type u_1} [group G] ⦃h k : set G⦄ (h' : h k) :

Subgroup closure of a set is monotone in its argument: if h ⊆ k, then closure h ≤ closure k.

theorem add_subgroup.closure_mono {G : Type u_1} [add_group G] ⦃h k : set G⦄ (h' : h k) :

Additive subgroup closure of a set is monotone in its argument: if h ⊆ k, then closure h ≤ closure k

@[simp]
theorem subgroup.closure_eq {G : Type u_1} [group G] (K : subgroup G) :

Closure of a subgroup K equals K.

@[simp]
theorem add_subgroup.closure_eq {G : Type u_1} [add_group G] (K : add_subgroup G) :

Additive closure of an additive subgroup K equals K

@[simp]
@[simp]
theorem subgroup.closure_empty {G : Type u_1} [group G] :
@[simp]
theorem subgroup.closure_univ {G : Type u_1} [group G] :
theorem subgroup.closure_union {G : Type u_1} [group G] (s t : set G) :
theorem subgroup.closure_Union {G : Type u_1} [group G] {ι : Sort u_2} (s : ι → set G) :
subgroup.closure (⋃ (i : ι), s i) = ⨆ (i : ι), subgroup.closure (s i)
theorem add_subgroup.closure_Union {G : Type u_1} [add_group G] {ι : Sort u_2} (s : ι → set G) :
add_subgroup.closure (⋃ (i : ι), s i) = ⨆ (i : ι), add_subgroup.closure (s i)
theorem subgroup.closure_eq_bot_iff (G : Type u_1) [group G] (S : set G) :
theorem add_subgroup.closure_eq_bot_iff (G : Type u_1) [add_group G] (S : set G) :
theorem add_subgroup.mem_closure_singleton {G : Type u_1} [add_group G] {x y : G} :
y add_subgroup.closure {x} ∃ (n : ), n x = y

The add_subgroup generated by an element of an add_group equals the set of natural number multiples of the element.

theorem subgroup.mem_closure_singleton {G : Type u_1} [group G] {x y : G} :
y subgroup.closure {x} ∃ (n : ), x ^ n = y

The subgroup generated by an element of a group equals the set of integer number powers of the element.

theorem subgroup.closure_singleton_one {G : Type u_1} [group G] :
@[simp]
theorem subgroup.inv_subset_closure {G : Type u_1} [group G] (S : set G) :
@[simp]
theorem add_subgroup.neg_subset_closure {G : Type u_1} [add_group G] (S : set G) :
@[simp]
theorem subgroup.closure_inv {G : Type u_1} [group G] (S : set G) :
@[simp]
theorem subgroup.closure_induction'' {G : Type u_1} [group G] {k : set G} {p : G → Prop} {x : G} (h : x subgroup.closure k) (Hk : ∀ (x : G), x kp x) (Hk_inv : ∀ (x : G), x kp x⁻¹) (H1 : p 1) (Hmul : ∀ (x y : G), p xp yp (x * y)) :
p x

An induction principle for closure membership. If p holds for 1 and all elements of k and their inverse, and is preserved under multiplication, then p holds for all elements of the closure of k.

theorem add_subgroup.closure_induction'' {G : Type u_1} [add_group G] {k : set G} {p : G → Prop} {x : G} (h : x add_subgroup.closure k) (Hk : ∀ (x : G), x kp x) (Hk_inv : ∀ (x : G), x kp (-x)) (H1 : p 0) (Hmul : ∀ (x y : G), p xp yp (x + y)) :
p x

An induction principle for additive closure membership. If p holds for 0 and all elements of k and their negation, and is preserved under addition, then p holds for all elements of the additive closure of k.

theorem add_subgroup.mem_supr_of_directed {G : Type u_1} [add_group G] {ι : Sort u_2} [hι : nonempty ι] {K : ι → add_subgroup G} (hK : directed has_le.le K) {x : G} :
x supr K ∃ (i : ι), x K i
theorem subgroup.mem_supr_of_directed {G : Type u_1} [group G] {ι : Sort u_2} [hι : nonempty ι] {K : ι → subgroup G} (hK : directed has_le.le K) {x : G} :
x supr K ∃ (i : ι), x K i
theorem subgroup.coe_supr_of_directed {G : Type u_1} [group G] {ι : Sort u_2} [nonempty ι] {S : ι → subgroup G} (hS : directed has_le.le S) :
(⨆ (i : ι), S i) = ⋃ (i : ι), (S i)
theorem add_subgroup.coe_supr_of_directed {G : Type u_1} [add_group G] {ι : Sort u_2} [nonempty ι] {S : ι → add_subgroup G} (hS : directed has_le.le S) :
(⨆ (i : ι), S i) = ⋃ (i : ι), (S i)
theorem add_subgroup.mem_Sup_of_directed_on {G : Type u_1} [add_group G] {K : set (add_subgroup G)} (Kne : K.nonempty) (hK : directed_on has_le.le K) {x : G} :
x Sup K ∃ (s : add_subgroup G) (H : s K), x s
theorem subgroup.mem_Sup_of_directed_on {G : Type u_1} [group G] {K : set (subgroup G)} (Kne : K.nonempty) (hK : directed_on has_le.le K) {x : G} :
x Sup K ∃ (s : subgroup G) (H : s K), x s
def subgroup.comap {G : Type u_1} [group G] {N : Type u_2} [group N] (f : G →* N) (H : subgroup N) :

The preimage of a subgroup along a monoid homomorphism is a subgroup.

Equations
def add_subgroup.comap {G : Type u_1} [add_group G] {N : Type u_2} [add_group N] (f : G →+ N) (H : add_subgroup N) :

The preimage of an add_subgroup along an add_monoid homomorphism is an add_subgroup.

Equations
@[simp]
theorem subgroup.coe_comap {G : Type u_1} [group G] {N : Type u_3} [group N] (K : subgroup N) (f : G →* N) :
@[simp]
theorem add_subgroup.coe_comap {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (K : add_subgroup N) (f : G →+ N) :
@[simp]
theorem add_subgroup.mem_comap {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {K : add_subgroup N} {f : G →+ N} {x : G} :
@[simp]
theorem subgroup.mem_comap {G : Type u_1} [group G] {N : Type u_3} [group N] {K : subgroup N} {f : G →* N} {x : G} :
theorem subgroup.comap_mono {G : Type u_1} [group G] {N : Type u_3} [group N] {f : G →* N} {K K' : subgroup N} :
theorem add_subgroup.comap_mono {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {f : G →+ N} {K K' : add_subgroup N} :
theorem add_subgroup.comap_comap {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {P : Type u_4} [add_group P] (K : add_subgroup P) (g : N →+ P) (f : G →+ N) :
theorem subgroup.comap_comap {G : Type u_1} [group G] {N : Type u_3} [group N] {P : Type u_4} [group P] (K : subgroup P) (g : N →* P) (f : G →* N) :
def add_subgroup.map {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) (H : add_subgroup G) :

The image of an add_subgroup along an add_monoid homomorphism is an add_subgroup.

Equations
def subgroup.map {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) (H : subgroup G) :

The image of a subgroup along a monoid homomorphism is a subgroup.

Equations
@[simp]
theorem add_subgroup.coe_map {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) (K : add_subgroup G) :
@[simp]
theorem subgroup.coe_map {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) (K : subgroup G) :
@[simp]
theorem add_subgroup.mem_map {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {f : G →+ N} {K : add_subgroup G} {y : N} :
y add_subgroup.map f K ∃ (x : G) (H : x K), f x = y
@[simp]
theorem subgroup.mem_map {G : Type u_1} [group G] {N : Type u_3} [group N] {f : G →* N} {K : subgroup G} {y : N} :
y subgroup.map f K ∃ (x : G) (H : x K), f x = y
theorem add_subgroup.mem_map_of_mem {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) {K : add_subgroup G} {x : G} (hx : x K) :
theorem subgroup.mem_map_of_mem {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) {K : subgroup G} {x : G} (hx : x K) :
theorem add_subgroup.apply_coe_mem_map {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) (K : add_subgroup G) (x : K) :
theorem subgroup.apply_coe_mem_map {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) (K : subgroup G) (x : K) :
theorem add_subgroup.map_mono {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {f : G →+ N} {K K' : add_subgroup G} :
theorem subgroup.map_mono {G : Type u_1} [group G] {N : Type u_3} [group N] {f : G →* N} {K K' : subgroup G} :
K K'subgroup.map f K subgroup.map f K'
@[simp]
theorem add_subgroup.map_id {G : Type u_1} [add_group G] (K : add_subgroup G) :
@[simp]
theorem subgroup.map_id {G : Type u_1} [group G] (K : subgroup G) :
theorem add_subgroup.map_map {G : Type u_1} [add_group G] (K : add_subgroup G) {N : Type u_3} [add_group N] {P : Type u_4} [add_group P] (g : N →+ P) (f : G →+ N) :
theorem subgroup.map_map {G : Type u_1} [group G] (K : subgroup G) {N : Type u_3} [group N] {P : Type u_4} [group P] (g : N →* P) (f : G →* N) :
theorem subgroup.mem_map_equiv {G : Type u_1} [group G] {N : Type u_3} [group N] {f : G ≃* N} {K : subgroup G} {x : N} :
theorem add_subgroup.mem_map_equiv {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {f : G ≃+ N} {K : add_subgroup G} {x : N} :
theorem subgroup.mem_map_iff_mem {G : Type u_1} [group G] {N : Type u_3} [group N] {f : G →* N} (hf : function.injective f) {K : subgroup G} {x : G} :
theorem add_subgroup.mem_map_iff_mem {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {f : G →+ N} (hf : function.injective f) {K : add_subgroup G} {x : G} :
theorem subgroup.map_equiv_eq_comap_symm {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G ≃* N) (K : subgroup G) :
theorem subgroup.comap_equiv_eq_map_symm {G : Type u_1} [group G] {N : Type u_3} [group N] (f : N ≃* G) (K : subgroup G) :
theorem add_subgroup.map_le_iff_le_comap {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {f : G →+ N} {K : add_subgroup G} {H : add_subgroup N} :
theorem subgroup.map_le_iff_le_comap {G : Type u_1} [group G] {N : Type u_3} [group N] {f : G →* N} {K : subgroup G} {H : subgroup N} :
theorem subgroup.gc_map_comap {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) :
theorem add_subgroup.gc_map_comap {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) :
theorem add_subgroup.map_sup {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (H K : add_subgroup G) (f : G →+ N) :
theorem subgroup.map_sup {G : Type u_1} [group G] {N : Type u_3} [group N] (H K : subgroup G) (f : G →* N) :
theorem add_subgroup.map_supr {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {ι : Sort u_2} (f : G →+ N) (s : ι → add_subgroup G) :
add_subgroup.map f (supr s) = ⨆ (i : ι), add_subgroup.map f (s i)
theorem subgroup.map_supr {G : Type u_1} [group G] {N : Type u_3} [group N] {ι : Sort u_2} (f : G →* N) (s : ι → subgroup G) :
subgroup.map f (supr s) = ⨆ (i : ι), subgroup.map f (s i)
theorem add_subgroup.comap_sup_comap_le {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (H K : add_subgroup N) (f : G →+ N) :
theorem subgroup.comap_sup_comap_le {G : Type u_1} [group G] {N : Type u_3} [group N] (H K : subgroup N) (f : G →* N) :
theorem add_subgroup.supr_comap_le {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {ι : Sort u_2} (f : G →+ N) (s : ι → add_subgroup N) :
(⨆ (i : ι), add_subgroup.comap f (s i)) add_subgroup.comap f (supr s)
theorem subgroup.supr_comap_le {G : Type u_1} [group G] {N : Type u_3} [group N] {ι : Sort u_2} (f : G →* N) (s : ι → subgroup N) :
(⨆ (i : ι), subgroup.comap f (s i)) subgroup.comap f (supr s)
theorem add_subgroup.comap_inf {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (H K : add_subgroup N) (f : G →+ N) :
theorem subgroup.comap_inf {G : Type u_1} [group G] {N : Type u_3} [group N] (H K : subgroup N) (f : G →* N) :
theorem subgroup.comap_infi {G : Type u_1} [group G] {N : Type u_3} [group N] {ι : Sort u_2} (f : G →* N) (s : ι → subgroup N) :
subgroup.comap f (infi s) = ⨅ (i : ι), subgroup.comap f (s i)
theorem add_subgroup.comap_infi {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {ι : Sort u_2} (f : G →+ N) (s : ι → add_subgroup N) :
add_subgroup.comap f (infi s) = ⨅ (i : ι), add_subgroup.comap f (s i)
theorem add_subgroup.map_inf_le {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (H K : add_subgroup G) (f : G →+ N) :
theorem subgroup.map_inf_le {G : Type u_1} [group G] {N : Type u_3} [group N] (H K : subgroup G) (f : G →* N) :
theorem subgroup.map_inf_eq {G : Type u_1} [group G] {N : Type u_3} [group N] (H K : subgroup G) (f : G →* N) (hf : function.injective f) :
theorem add_subgroup.map_inf_eq {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (H K : add_subgroup G) (f : G →+ N) (hf : function.injective f) :
@[simp]
theorem add_subgroup.map_bot {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) :
@[simp]
theorem subgroup.map_bot {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) :
@[simp]
theorem subgroup.comap_top {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) :
@[simp]
theorem add_subgroup.comap_top {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) :
@[simp]
theorem subgroup.comap_subtype_inf_left {G : Type u_1} [group G] {H K : subgroup G} :
@[simp]
theorem subgroup.comap_subtype_inf_right {G : Type u_1} [group G] {H K : subgroup G} :
def subgroup.comap_subtype_equiv_of_le {G : Type u_1} [group G] {H K : subgroup G} (h : H K) :

If H ≤ K, then H as a subgroup of K is isomorphic to H.

Equations

If H ≤ K, then H as a subgroup of K is isomorphic to H.

Equations
@[simp]
@[simp]
def add_subgroup.add_subgroup_of {G : Type u_1} [add_group G] (H K : add_subgroup G) :

For any subgroups H and K, view H ⊓ K as a subgroup of K.

Equations
def subgroup.subgroup_of {G : Type u_1} [group G] (H K : subgroup G) :

For any subgroups H and K, view H ⊓ K as a subgroup of K.

Equations
theorem subgroup.coe_subgroup_of {G : Type u_1} [group G] (H K : subgroup G) :
theorem subgroup.mem_subgroup_of {G : Type u_1} [group G] {H K : subgroup G} {h : K} :
theorem add_subgroup.mem_add_subgroup_of {G : Type u_1} [add_group G] {H K : add_subgroup G} {h : K} :
theorem subgroup.subgroup_of_map_subtype {G : Type u_1} [group G] (H K : subgroup G) :
@[simp]
@[simp]
theorem subgroup.bot_subgroup_of {G : Type u_1} [group G] (H : subgroup G) :
@[simp]
theorem subgroup.top_subgroup_of {G : Type u_1} [group G] (H : subgroup G) :
@[simp]
theorem subgroup.subgroup_of_bot_eq_bot {G : Type u_1} [group G] (H : subgroup G) :
theorem subgroup.subgroup_of_bot_eq_top {G : Type u_1} [group G] (H : subgroup G) :
@[simp]
theorem subgroup.subgroup_of_self {G : Type u_1} [group G] (H : subgroup G) :
@[simp]
theorem add_subgroup.add_subgroup_of_self {G : Type u_1} [add_group G] (H : add_subgroup G) :
def add_subgroup.prod {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (H : add_subgroup G) (K : add_subgroup N) :

Given add_subgroups H, K of add_groups A, B respectively, H × K as an add_subgroup of A × B.

Equations
def subgroup.prod {G : Type u_1} [group G] {N : Type u_3} [group N] (H : subgroup G) (K : subgroup N) :
subgroup (G × N)

Given subgroups H, K of groups G, N respectively, H × K as a subgroup of G × N.

Equations
theorem subgroup.coe_prod {G : Type u_1} [group G] {N : Type u_3} [group N] (H : subgroup G) (K : subgroup N) :
(H.prod K) = H.prod K
theorem add_subgroup.coe_prod {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (H : add_subgroup G) (K : add_subgroup N) :
(H.prod K) = H.prod K
theorem add_subgroup.mem_prod {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {H : add_subgroup G} {K : add_subgroup N} {p : G × N} :
p H.prod K p.fst H p.snd K
theorem subgroup.mem_prod {G : Type u_1} [group G] {N : Type u_3} [group N] {H : subgroup G} {K : subgroup N} {p : G × N} :
p H.prod K p.fst H p.snd K
theorem subgroup.prod_mono {G : Type u_1} [group G] {N : Type u_3} [group N] :
theorem subgroup.prod_mono_right {G : Type u_1} [group G] {N : Type u_3} [group N] (K : subgroup G) :
monotone (λ (t : subgroup N), K.prod t)
theorem add_subgroup.prod_mono_right {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (K : add_subgroup G) :
monotone (λ (t : add_subgroup N), K.prod t)
theorem add_subgroup.prod_mono_left {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (H : add_subgroup N) :
monotone (λ (K : add_subgroup G), K.prod H)
theorem subgroup.prod_mono_left {G : Type u_1} [group G] {N : Type u_3} [group N] (H : subgroup N) :
monotone (λ (K : subgroup G), K.prod H)
theorem add_subgroup.prod_top {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (K : add_subgroup G) :
theorem subgroup.prod_top {G : Type u_1} [group G] {N : Type u_3} [group N] (K : subgroup G) :
theorem add_subgroup.top_prod {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (H : add_subgroup N) :
theorem subgroup.top_prod {G : Type u_1} [group G] {N : Type u_3} [group N] (H : subgroup N) :
@[simp]
theorem subgroup.top_prod_top {G : Type u_1} [group G] {N : Type u_3} [group N] :
@[simp]
theorem add_subgroup.top_prod_top {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] :
theorem subgroup.bot_prod_bot {G : Type u_1} [group G] {N : Type u_3} [group N] :
theorem add_subgroup.bot_sum_bot {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] :
def subgroup.prod_equiv {G : Type u_1} [group G] {N : Type u_3} [group N] (H : subgroup G) (K : subgroup N) :

Product of subgroups is isomorphic to their product as groups.

Equations
def add_subgroup.prod_equiv {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (H : add_subgroup G) (K : add_subgroup N) :

Product of additive subgroups is isomorphic to their product as additive groups

Equations
@[protected, instance]
def subgroup.normal_of_comm {G : Type u_1} [comm_group G] (H : subgroup G) :
@[protected, instance]
def add_subgroup.normal_of_comm {G : Type u_1} [add_comm_group G] (H : add_subgroup G) :
theorem add_subgroup.normal.mem_comm {G : Type u_1} [add_group G] {H : add_subgroup G} (nH : H.normal) {a b : G} (h : a + b H) :
b + a H
theorem subgroup.normal.mem_comm {G : Type u_1} [group G] {H : subgroup G} (nH : H.normal) {a b : G} (h : a * b H) :
b * a H
theorem subgroup.normal.mem_comm_iff {G : Type u_1} [group G] {H : subgroup G} (nH : H.normal) {a b : G} :
a * b H b * a H
theorem add_subgroup.normal.mem_comm_iff {G : Type u_1} [add_group G] {H : add_subgroup G} (nH : H.normal) {a b : G} :
a + b H b + a H
@[class]
structure subgroup.characteristic {G : Type u_1} [group G] (H : subgroup G) :
Prop

A subgroup is characteristic if it is fixed by all automorphisms. Several equivalent conditions are provided by lemmas of the form characteristic.iff...

Instances
@[protected, instance]
def subgroup.normal_of_characteristic {G : Type u_1} [group G] (H : subgroup G) [h : H.characteristic] :
@[class]
structure add_subgroup.characteristic {A : Type u_2} [add_group A] (H : add_subgroup A) :
Prop

A add_subgroup is characteristic if it is fixed by all automorphisms. Several equivalent conditions are provided by lemmas of the form characteristic.iff...

Instances
@[protected, instance]
theorem subgroup.characteristic_iff_comap_eq {G : Type u_1} [group G] {H : subgroup G} :
theorem subgroup.characteristic_iff_comap_le {G : Type u_1} [group G] {H : subgroup G} :
theorem subgroup.characteristic_iff_le_comap {G : Type u_1} [group G] {H : subgroup G} :
theorem subgroup.characteristic_iff_map_eq {G : Type u_1} [group G] {H : subgroup G} :
theorem subgroup.characteristic_iff_map_le {G : Type u_1} [group G] {H : subgroup G} :
theorem subgroup.characteristic_iff_le_map {G : Type u_1} [group G] {H : subgroup G} :
@[protected, instance]
@[protected, instance]
@[protected, instance]
@[protected, instance]
def add_subgroup.center (G : Type u_1) [add_group G] :

The center of an additive group G is the set of elements that commute with everything in G

Equations
def subgroup.center (G : Type u_1) [group G] :

The center of a group G is the set of elements that commute with everything in G

Equations
theorem subgroup.coe_center (G : Type u_1) [group G] :
theorem subgroup.mem_center_iff {G : Type u_1} [group G] {z : G} :
z subgroup.center G ∀ (g : G), g * z = z * g
theorem add_subgroup.mem_center_iff {G : Type u_1} [add_group G] {z : G} :
z add_subgroup.center G ∀ (g : G), g + z = z + g
@[protected, instance]
def subgroup.decidable_mem_center {G : Type u_1} [group G] [decidable_eq G] [fintype G] :
decidable_pred (λ (_x : G), _x subgroup.center G)
Equations
@[protected, instance]
@[protected, instance]
def add_subgroup.normalizer {G : Type u_1} [add_group G] (H : add_subgroup G) :

The normalizer of H is the largest subgroup of G inside which H is normal.

Equations
def subgroup.normalizer {G : Type u_1} [group G] (H : subgroup G) :

The normalizer of H is the largest subgroup of G inside which H is normal.

Equations
def subgroup.set_normalizer {G : Type u_1} [group G] (S : set G) :

The set_normalizer of S is the subgroup of G whose elements satisfy g*S*g⁻¹=S

Equations
def add_subgroup.set_normalizer {G : Type u_1} [add_group G] (S : set G) :

The set_normalizer of S is the subgroup of G whose elements satisfy g+S-g=S.

Equations
theorem subgroup.mem_normalizer_fintype {G : Type u_1} [group G] {S : set G} [fintype S] {x : G} (h : ∀ (n : G), n S(x * n) * x⁻¹ S) :
theorem add_subgroup.mem_normalizer_iff {G : Type u_1} [add_group G] {H : add_subgroup G} {g : G} :
g H.normalizer ∀ (n : G), n H g + n + -g H
theorem subgroup.mem_normalizer_iff {G : Type u_1} [group G] {H : subgroup G} {g : G} :
g H.normalizer ∀ (n : G), n H (g * n) * g⁻¹ H
theorem add_subgroup.le_normalizer {G : Type u_1} [add_group G] {H : add_subgroup G} :
theorem subgroup.le_normalizer {G : Type u_1} [group G] {H : subgroup G} :
@[protected, instance]
@[protected, instance]
theorem subgroup.normalizer_eq_top {G : Type u_1} [group G] {H : subgroup G} :
theorem subgroup.le_normalizer_of_normal {G : Type u_1} [group G] {H K : subgroup G} [hK : (subgroup.comap K.subtype H).normal] (HK : H K) :
theorem add_subgroup.le_normalizer_of_normal {G : Type u_1} [add_group G] {H K : add_subgroup G} [hK : (add_subgroup.comap K.subtype H).normal] (HK : H K) :
theorem subgroup.le_normalizer_comap {G : Type u_1} [group G] {H : subgroup G} {N : Type u_3} [group N] (f : N →* G) :

The preimage of the normalizer is contained in the normalizer of the preimage.

theorem add_subgroup.le_normalizer_comap {G : Type u_1} [add_group G] {H : add_subgroup G} {N : Type u_3} [add_group N] (f : N →+ G) :

The preimage of the normalizer is contained in the normalizer of the preimage.

theorem subgroup.le_normalizer_map {G : Type u_1} [group G] {H : subgroup G} {N : Type u_3} [group N] (f : G →* N) :

The image of the normalizer is contained in the normalizer of the image.

theorem add_subgroup.le_normalizer_map {G : Type u_1} [add_group G] {H : add_subgroup G} {N : Type u_3} [add_group N] (f : G →+ N) :

The image of the normalizer is contained in the normalizer of the image.

@[class]
structure subgroup.is_commutative {G : Type u_1} [group G] (H : subgroup G) :
Prop

Commutivity of a subgroup

Instances
@[class]
structure add_subgroup.is_commutative {A : Type u_2} [add_group A] (H : add_subgroup A) :
Prop

Commutivity of an additive subgroup

@[protected, instance]
def subgroup.is_commutative.comm_group {G : Type u_1} [group G] (H : subgroup G) [h : H.is_commutative] :

A commutative subgroup is commutative

Equations
@[protected, instance]
def group.conjugates_of_set {G : Type u_1} [group G] (s : set G) :
set G

Given a set s, conjugates_of_set s is the set of all conjugates of the elements of s.

Equations
theorem group.mem_conjugates_of_set_iff {G : Type u_1} [group G] {s : set G} {x : G} :
x group.conjugates_of_set s ∃ (a : G) (H : a s), is_conj a x
theorem group.subset_conjugates_of_set {G : Type u_1} [group G] {s : set G} :
theorem group.conjugates_of_set_mono {G : Type u_1} [group G] {s t : set G} (h : s t) :
theorem group.conjugates_subset_normal {G : Type u_1} [group G] {N : subgroup G} [tn : N.normal] {a : G} (h : a N) :
theorem group.conjugates_of_set_subset {G : Type u_1} [group G] {s : set G} {N : subgroup G} [N.normal] (h : s N) :
theorem group.conj_mem_conjugates_of_set {G : Type u_1} [group G] {s : set G} {x c : G} :

The set of conjugates of s is closed under conjugation.

def subgroup.normal_closure {G : Type u_1} [group G] (s : set G) :

The normal closure of a set s is the subgroup closure of all the conjugates of elements of s. It is the smallest normal subgroup containing s.

Equations
theorem subgroup.subset_normal_closure {G : Type u_1} [group G] {s : set G} :
@[protected, instance]

The normal closure of s is a normal subgroup.

theorem subgroup.normal_closure_le_normal {G : Type u_1} [group G] {s : set G} {N : subgroup G} [N.normal] (h : s N) :

The normal closure of s is the smallest normal subgroup containing s.

theorem subgroup.normal_closure_subset_iff {G : Type u_1} [group G] {s : set G} {N : subgroup G} [N.normal] :
theorem subgroup.normal_closure_mono {G : Type u_1} [group G] {s t : set G} (h : s t) :
theorem subgroup.normal_closure_eq_infi {G : Type u_1} [group G] {s : set G} :
subgroup.normal_closure s = ⨅ (N : subgroup G) (_x : N.normal) (hs : s N), N
@[simp]
theorem subgroup.normal_closure_eq_self {G : Type u_1} [group G] (H : subgroup G) [H.normal] :
def subgroup.normal_core {G : Type u_1} [group G] (H : subgroup G) :

The normal core of a subgroup H is the largest normal subgroup of G contained in H, as shown by subgroup.normal_core_eq_supr.

Equations
theorem subgroup.normal_core_le {G : Type u_1} [group G] (H : subgroup G) :
@[protected, instance]
def subgroup.normal_core_normal {G : Type u_1} [group G] (H : subgroup G) :
theorem subgroup.normal_le_normal_core {G : Type u_1} [group G] {H N : subgroup G} [hN : N.normal] :
theorem subgroup.normal_core_mono {G : Type u_1} [group G] {H K : subgroup G} (h : H K) :
theorem subgroup.normal_core_eq_supr {G : Type u_1} [group G] (H : subgroup G) :
H.normal_core = ⨆ (N : subgroup G) (_x : N.normal) (hs : N H), N
@[simp]
theorem subgroup.normal_core_eq_self {G : Type u_1} [group G] (H : subgroup G) [H.normal] :
@[simp]
def add_monoid_hom.range {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) :

The range of an add_monoid_hom from an add_group is an add_subgroup.

Equations
def monoid_hom.range {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) :

The range of a monoid homomorphism from a group is a subgroup.

Equations
@[protected, instance]
def add_monoid_hom.decidable_mem_range {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) [fintype G] [decidable_eq N] :
decidable_pred (λ (_x : N), _x f.range)
Equations
@[protected, instance]
def monoid_hom.decidable_mem_range {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) [fintype G] [decidable_eq N] :
decidable_pred (λ (_x : N), _x f.range)
Equations
@[simp]
theorem add_monoid_hom.coe_range {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) :
@[simp]
theorem monoid_hom.coe_range {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) :
@[simp]
theorem monoid_hom.mem_range {G : Type u_1} [group G] {N : Type u_3} [group N] {f : G →* N} {y : N} :
y f.range ∃ (x : G), f x = y
@[simp]
theorem add_monoid_hom.mem_range {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {f : G →+ N} {y : N} :
y f.range ∃ (x : G), f x = y
theorem monoid_hom.range_eq_map {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) :
theorem add_monoid_hom.range_eq_map {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) :
def monoid_hom.range_restrict {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) :

The canonical surjective group homomorphism G →* f(G) induced by a group homomorphism G →* N.

Equations
def add_monoid_hom.range_restrict {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) :

The canonical surjective add_group homomorphism G →+ f(G) induced by a group homomorphism G →+ N.

Equations
@[simp]
theorem monoid_hom.coe_range_restrict {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) (g : G) :
@[simp]
theorem add_monoid_hom.coe_range_restrict {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) (g : G) :
theorem monoid_hom.range_restrict_surjective {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) :
theorem add_monoid_hom.range_restrict_surjective {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) :
theorem monoid_hom.map_range {G : Type u_1} [group G] {N : Type u_3} {P : Type u_4} [group N] [group P] (g : N →* P) (f : G →* N) :
theorem add_monoid_hom.map_range {G : Type u_1} [add_group G] {N : Type u_3} {P : Type u_4} [add_group N] [add_group P] (g : N →+ P) (f : G →+ N) :
theorem monoid_hom.range_top_iff_surjective {G : Type u_1} [group G] {N : Type u_2} [group N] {f : G →* N} :
theorem add_monoid_hom.range_top_iff_surjective {G : Type u_1} [add_group G] {N : Type u_2} [add_group N] {f : G →+ N} :
theorem monoid_hom.range_top_of_surjective {G : Type u_1} [group G] {N : Type u_2} [group N] (f : G →* N) (hf : function.surjective f) :

The range of a surjective monoid homomorphism is the whole of the codomain.

theorem add_monoid_hom.range_top_of_surjective {G : Type u_1} [add_group G] {N : Type u_2} [add_group N] (f : G →+ N) (hf : function.surjective f) :

The range of a surjective add_monoid homomorphism is the whole of the codomain.

@[simp]
theorem add_subgroup.subtype_range {G : Type u_1} [add_group G] (H : add_subgroup G) :
@[simp]
theorem subgroup.subtype_range {G : Type u_1} [group G] (H : subgroup G) :
@[simp]
theorem add_subgroup.inclusion_range {G : Type u_1} [add_group G] {H K : add_subgroup G} (h_le : H K) :
@[simp]
theorem subgroup.inclusion_range {G : Type u_1} [group G] {H K : subgroup G} (h_le : H K) :
def add_monoid_hom.restrict {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) (H : add_subgroup G) :

Restriction of an add_group hom to an add_subgroup of the domain.

Equations
def monoid_hom.restrict {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) (H : subgroup G) :

Restriction of a group hom to a subgroup of the domain.

Equations
@[simp]
theorem add_monoid_hom.restrict_apply {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {H : add_subgroup G} (f : G →+ N) (x : H) :
(f.restrict H) x = f x
@[simp]
theorem monoid_hom.restrict_apply {G : Type u_1} [group G] {N : Type u_3} [group N] {H : subgroup G} (f : G →* N) (x : H) :
(f.restrict H) x = f x
def add_monoid_hom.cod_restrict {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) (S : add_subgroup N) (h : ∀ (x : G), f x S) :

Restriction of an add_group hom to an add_subgroup of the codomain.

Equations
def monoid_hom.cod_restrict {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) (S : subgroup N) (h : ∀ (x : G), f x S) :

Restriction of a group hom to a subgroup of the codomain.

Equations
@[simp]
theorem monoid_hom.cod_restrict_apply {G : Type u_1} [group G] {N : Type u_2} [group N] (f : G →* N) (S : subgroup N) (h : ∀ (x : G), f x S) {x : G} :
(f.cod_restrict S h) x = f x, _⟩
@[simp]
theorem add_monoid_hom.cod_restrict_apply {G : Type u_1} [add_group G] {N : Type u_2} [add_group N] (f : G →+ N) (S : add_subgroup N) (h : ∀ (x : G), f x S) {x : G} :
(f.cod_restrict S h) x = f x, _⟩
theorem monoid_hom.subgroup_of_range_eq_of_le {G₁ : Type u_1} {G₂ : Type u_2} [group G₁] [group G₂] {K : subgroup G₂} (f : G₁ →* G₂) (h : f.range K) :
theorem add_monoid_hom.add_subgroup_of_range_eq_of_le {G₁ : Type u_1} {G₂ : Type u_2} [add_group G₁] [add_group G₂] {K : add_subgroup G₂} (f : G₁ →+ G₂) (h : f.range K) :
def monoid_hom.of_left_inverse {G : Type u_1} [group G] {N : Type u_3} [group N] {f : G →* N} {g : N →* G} (h : function.left_inverse g f) :

Computable alternative to monoid_hom.of_injective.

Equations
def add_monoid_hom.of_left_inverse {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {f : G →+ N} {g : N →+ G} (h : function.left_inverse g f) :

Computable alternative to add_monoid_hom.of_injective.

Equations
@[simp]
theorem monoid_hom.of_left_inverse_apply {G : Type u_1} [group G] {N : Type u_3} [group N] {f : G →* N} {g : N →* G} (h : function.left_inverse g f) (x : G) :
@[simp]
theorem add_monoid_hom.of_left_inverse_apply {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {f : G →+ N} {g : N →+ G} (h : function.left_inverse g f) (x : G) :
@[simp]
theorem add_monoid_hom.of_left_inverse_symm_apply {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {f : G →+ N} {g : N →+ G} (h : function.left_inverse g f) (x : (f.range)) :
@[simp]
theorem monoid_hom.of_left_inverse_symm_apply {G : Type u_1} [group G] {N : Type u_3} [group N] {f : G →* N} {g : N →* G} (h : function.left_inverse g f) (x : (f.range)) :
noncomputable def monoid_hom.of_injective {G : Type u_1} [group G] {N : Type u_3} [group N] {f : G →* N} (hf : function.injective f) :

The range of an injective group homomorphism is isomorphic to its domain.

Equations
noncomputable def add_monoid_hom.of_injective {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {f : G →+ N} (hf : function.injective f) :

The range of an injective additive group homomorphism is isomorphic to its domain.

Equations
theorem monoid_hom.of_injective_apply {G : Type u_1} [group G] {N : Type u_3} [group N] {f : G →* N} (hf : function.injective f) {x : G} :
theorem add_monoid_hom.of_injective_apply {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {f : G →+ N} (hf : function.injective f) {x : G} :
def monoid_hom.ker {G : Type u_1} [group G] {M : Type u_5} [mul_one_class M] (f : G →* M) :

The multiplicative kernel of a monoid homomorphism is the subgroup of elements x : G such that f x = 1

Equations
def add_monoid_hom.ker {G : Type u_1} [add_group G] {M : Type u_5} [add_zero_class M] (f : G →+ M) :

The additive kernel of an add_monoid homomorphism is the add_subgroup of elements such that f x = 0

Equations
theorem monoid_hom.mem_ker {G : Type u_1} [group G] {M : Type u_5} [mul_one_class M] (f : G →* M) {x : G} :
x f.ker f x = 1
theorem add_monoid_hom.mem_ker {G : Type u_1} [add_group G] {M : Type u_5} [add_zero_class M] (f : G →+ M) {x : G} :
x f.ker f x = 0
theorem add_monoid_hom.coe_ker {G : Type u_1} [add_group G] {M : Type u_5} [add_zero_class M] (f : G →+ M) :
(f.ker) = f ⁻¹' {0}
theorem monoid_hom.coe_ker {G : Type u_1} [group G] {M : Type u_5} [mul_one_class M] (f : G →* M) :
(f.ker) = f ⁻¹' {1}
theorem add_monoid_hom.eq_iff {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) {x y : G} :
f x = f y -y + x f.ker
theorem monoid_hom.eq_iff {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) {x y : G} :
f x = f y y⁻¹ * x f.ker
@[protected, instance]
def monoid_hom.decidable_mem_ker {G : Type u_1} [group G] {M : Type u_5} [mul_one_class M] [decidable_eq M] (f : G →* M) :
decidable_pred (λ (_x : G), _x f.ker)
Equations
@[protected, instance]
def add_monoid_hom.decidable_mem_ker {G : Type u_1} [add_group G] {M : Type u_5} [add_zero_class M] [decidable_eq M] (f : G →+ M) :
decidable_pred (λ (_x : G), _x f.ker)
Equations
theorem add_monoid_hom.comap_ker {G : Type u_1} [add_group G] {N : Type u_3} {P : Type u_4} [add_group N] [add_group P] (g : N →+ P) (f : G →+ N) :
theorem monoid_hom.comap_ker {G : Type u_1} [group G] {N : Type u_3} {P : Type u_4} [group N] [group P] (g : N →* P) (f : G →* N) :
@[simp]
theorem monoid_hom.comap_bot {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) :
@[simp]
theorem add_monoid_hom.comap_bot {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) :
theorem add_monoid_hom.range_restrict_ker {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) :
theorem monoid_hom.range_restrict_ker {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) :
@[simp]
theorem monoid_hom.ker_one {G : Type u_1} [group G] {M : Type u_5} [mul_one_class M] :
@[simp]
theorem add_monoid_hom.ker_zero {G : Type u_1} [add_group G] {M : Type u_5} [add_zero_class M] :
theorem monoid_hom.ker_eq_bot_iff {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) :
theorem add_monoid_hom.ker_eq_bot_iff {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) :
@[simp]
theorem subgroup.ker_subtype {G : Type u_1} [group G] (H : subgroup G) :
@[simp]
theorem add_subgroup.ker_subtype {G : Type u_1} [add_group G] (H : add_subgroup G) :
@[simp]
theorem subgroup.ker_inclusion {G : Type u_1} [group G] {H K : subgroup G} (h : H K) :
@[simp]
theorem add_subgroup.ker_inclusion {G : Type u_1} [add_group G] {H K : add_subgroup G} (h : H K) :
theorem add_monoid_hom.sum_map_comap_sum {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {G' : Type u_2} {N' : Type u_4} [add_group G'] [add_group N'] (f : G →+ N) (g : G' →+ N') (S : add_subgroup N) (S' : add_subgroup N') :
theorem monoid_hom.prod_map_comap_prod {G : Type u_1} [group G] {N : Type u_3} [group N] {G' : Type u_2} {N' : Type u_4} [group G'] [group N'] (f : G →* N) (g : G' →* N') (S : subgroup N) (S' : subgroup N') :
theorem monoid_hom.ker_prod_map {G : Type u_1} [group G] {N : Type u_3} [group N] {G' : Type u_2} {N' : Type u_4} [group G'] [group N'] (f : G →* N) (g : G' →* N') :
(f.prod_map g).ker = f.ker.prod g.ker
theorem add_monoid_hom.ker_sum_map {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {G' : Type u_2} {N' : Type u_4} [add_group G'] [add_group N'] (f : G →+ N) (g : G' →+ N') :
(f.prod_map g).ker = f.ker.prod g.ker
def monoid_hom.eq_locus {G : Type u_1} [group G] {N : Type u_3} [group N] (f g : G →* N) :

The subgroup of elements x : G such that f x = g x

Equations
def add_monoid_hom.eq_locus {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f g : G →+ N) :

The additive subgroup of elements x : G such that f x = g x

Equations
theorem add_monoid_hom.eq_on_closure {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {f g : G →+ N} {s : set G} (h : set.eq_on f g s) :
theorem monoid_hom.eq_on_closure {G : Type u_1} [group G] {N : Type u_3} [group N] {f g : G →* N} {s : set G} (h : set.eq_on f g s) :

If two monoid homomorphisms are equal on a set, then they are equal on its subgroup closure.

theorem add_monoid_hom.eq_of_eq_on_top {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {f g : G →+ N} (h : set.eq_on f g ) :
f = g
theorem monoid_hom.eq_of_eq_on_top {G : Type u_1} [group G] {N : Type u_3} [group N] {f g : G →* N} (h : set.eq_on f g ) :
f = g
theorem add_monoid_hom.eq_of_eq_on_dense {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {s : set G} (hs : add_subgroup.closure s = ) {f g : G →+ N} (h : set.eq_on f g s) :
f = g
theorem monoid_hom.eq_of_eq_on_dense {G : Type u_1} [group G] {N : Type u_3} [group N] {s : set G} (hs : subgroup.closure s = ) {f g : G →* N} (h : set.eq_on f g s) :
f = g
theorem monoid_hom.gclosure_preimage_le {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) (s : set N) :
theorem add_monoid_hom.gclosure_preimage_le {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) (s : set N) :
theorem add_monoid_hom.map_closure {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) (s : set G) :

The image under an add_monoid hom of the add_subgroup generated by a set equals the add_subgroup generated by the image of the set.

theorem monoid_hom.map_closure {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) (s : set G) :

The image under a monoid homomorphism of the subgroup generated by a set equals the subgroup generated by the image of the set.

@[protected, instance]
def add_monoid_hom.fintype_mrange {M : Type u_1} {N : Type u_2} [add_monoid M] [add_monoid N] [fintype M] [decidable_eq N] (f : M →+ N) :

The range of a finite additive monoid under an additive monoid homomorphism is finite.

Note: this instance can form a diamond with subtype.fintype or subgroup.fintype in the presence of fintype N.

Equations
@[protected, instance]
def monoid_hom.fintype_mrange {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] [fintype M] [decidable_eq N] (f : M →* N) :

The range of a finite monoid under a monoid homomorphism is finite. Note: this instance can form a diamond with subtype.fintype in the presence of fintype N.

Equations
@[protected, instance]
def monoid_hom.fintype_range {G : Type u_1} [group G] {N : Type u_3} [group N] [fintype G] [decidable_eq N] (f : G →* N) :

The range of a finite group under a group homomorphism is finite.

Note: this instance can form a diamond with subtype.fintype or subgroup.fintype in the presence of fintype N.

Equations
@[protected, instance]
def add_monoid_hom.fintype_range {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] [fintype G] [decidable_eq N] (f : G →+ N) :

The range of a finite additive group under an additive group homomorphism is finite.

Note: this instance can form a diamond with subtype.fintype or subgroup.fintype in the presence of fintype N.

Equations
theorem subgroup.map_eq_bot_iff {G : Type u_1} [group G] {N : Type u_3} [group N] (H : subgroup G) {f : G →* N} :
theorem add_subgroup.map_eq_bot_iff {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (H : add_subgroup G) {f : G →+ N} :
theorem subgroup.map_eq_bot_iff_of_injective {G : Type u_1} [group G] {N : Type u_3} [group N] (H : subgroup G) {f : G →* N} (hf : function.injective f) :
theorem add_subgroup.map_eq_bot_iff_of_injective {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (H : add_subgroup G) {f : G →+ N} (hf : function.injective f) :
theorem add_subgroup.map_le_range {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) (H : add_subgroup G) :
theorem subgroup.map_le_range {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) (H : subgroup G) :
theorem subgroup.map_subtype_le {G : Type u_1} [group G] {H : subgroup G} (K : subgroup H) :
theorem subgroup.ker_le_comap {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) (H : subgroup N) :
theorem add_subgroup.ker_le_comap {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) (H : add_subgroup N) :
theorem add_subgroup.map_comap_le {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) (H : add_subgroup N) :
theorem subgroup.map_comap_le {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) (H : subgroup N) :
theorem subgroup.le_comap_map {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) (H : subgroup G) :
theorem add_subgroup.le_comap_map {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) (H : add_subgroup G) :
theorem add_subgroup.map_comap_eq {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) (H : add_subgroup N) :
theorem subgroup.map_comap_eq {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) (H : subgroup N) :
theorem add_subgroup.comap_map_eq {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) (H : add_subgroup G) :
theorem subgroup.comap_map_eq {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) (H : subgroup G) :
theorem add_subgroup.map_comap_eq_self {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {f : G →+ N} {H : add_subgroup N} (h : H f.range) :
theorem subgroup.map_comap_eq_self {G : Type u_1} [group G] {N : Type u_3} [group N] {f : G →* N} {H : subgroup N} (h : H f.range) :
theorem add_subgroup.map_comap_eq_self_of_surjective {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {f : G →+ N} (h : function.surjective f) (H : add_subgroup N) :
theorem subgroup.map_comap_eq_self_of_surjective {G : Type u_1} [group G] {N : Type u_3} [group N] {f : G →* N} (h : function.surjective f) (H : subgroup N) :
theorem subgroup.comap_injective {G : Type u_1} [group G] {N : Type u_3} [group N] {f : G →* N} (h : function.surjective f) :
theorem add_subgroup.comap_injective {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {f : G →+ N} (h : function.surjective f) :
theorem subgroup.comap_map_eq_self {G : Type u_1} [group G] {N : Type u_3} [group N] {f : G →* N} {H : subgroup G} (h : f.ker H) :
theorem add_subgroup.comap_map_eq_self {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {f : G →+ N} {H : add_subgroup G} (h : f.ker H) :
theorem subgroup.comap_map_eq_self_of_injective {G : Type u_1} [group G] {N : Type u_3} [group N] {f : G →* N} (h : function.injective f) (H : subgroup G) :
theorem add_subgroup.comap_map_eq_self_of_injective {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {f : G →+ N} (h : function.injective f) (H : add_subgroup G) :
theorem add_subgroup.map_injective {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {f : G →+ N} (h : function.injective f) :
theorem subgroup.map_injective {G : Type u_1} [group G] {N : Type u_3} [group N] {f : G →* N} (h : function.injective f) :
theorem subgroup.map_eq_comap_of_inverse {G : Type u_1} [group G] {N : Type u_3} [group N] {f : G →* N} {g : N →* G} (hl : function.left_inverse g f) (hr : function.right_inverse g f) (H : subgroup G) :
theorem add_subgroup.map_eq_comap_of_inverse {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {f : G →+ N} {g : N →+ G} (hl : function.left_inverse g f) (hr : function.right_inverse g f) (H : add_subgroup G) :
theorem add_subgroup.map_injective_of_ker_le {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) {H K : add_subgroup G} (hH : f.ker H) (hK : f.ker K) (hf : add_subgroup.map f H = add_subgroup.map f K) :
H = K
theorem subgroup.map_injective_of_ker_le {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) {H K : subgroup G} (hH : f.ker H) (hK : f.ker K) (hf : subgroup.map f H = subgroup.map f K) :
H = K

Given f(A) = f(B), ker f ≤ A, and ker f ≤ B, deduce that A = B

theorem subgroup.comap_sup_eq_of_le_range {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) {H K : subgroup N} (hH : H f.range) (hK : K f.range) :
theorem add_subgroup.comap_sup_eq_of_le_range {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) {H K : add_subgroup N} (hH : H f.range) (hK : K f.range) :
theorem add_subgroup.comap_sup_eq {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) (H K : add_subgroup N) (hf : function.surjective f) :
theorem subgroup.comap_sup_eq {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) (H K : subgroup N) (hf : function.surjective f) :
theorem subgroup.sup_subgroup_of_eq {G : Type u_1} [group G] {H K L : subgroup G} (hH : H L) (hK : K L) :
theorem add_subgroup.sup_add_subgroup_of_eq {G : Type u_1} [add_group G] {H K L : add_subgroup G} (hH : H L) (hK : K L) :
noncomputable def subgroup.equiv_map_of_injective {G : Type u_1} [group G] {N : Type u_3} [group N] (H : subgroup G) (f : G →* N) (hf : function.injective f) :

A subgroup is isomorphic to its image under an injective function

Equations
noncomputable def add_subgroup.equiv_map_of_injective {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (H : add_subgroup G) (f : G →+ N) (hf : function.injective f) :

An additive subgroup is isomorphic to its image under an injective function

Equations
@[simp]
theorem subgroup.coe_equiv_map_of_injective_apply {G : Type u_1} [group G] {N : Type u_3} [group N] (H : subgroup G) (f : G →* N) (hf : function.injective f) (h : H) :
@[simp]
theorem add_subgroup.coe_equiv_map_of_injective_apply {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (H : add_subgroup G) (f : G →+ N) (hf : function.injective f) (h : H) :

The preimage of the normalizer is equal to the normalizer of the preimage of a surjective function.

theorem subgroup.comap_normalizer_eq_of_surjective {G : Type u_1} [group G] {N : Type u_3} [group N] (H : subgroup G) {f : N →* G} (hf : function.surjective f) :

The preimage of the normalizer is equal to the normalizer of the preimage of a surjective function.

theorem subgroup.map_equiv_normalizer_eq {G : Type u_1} [group G] {N : Type u_3} [group N] (H : subgroup G) (f : G ≃* N) :

The image of the normalizer is equal to the normalizer of the image of an isomorphism.

The image of the normalizer is equal to the normalizer of the image of an isomorphism.

theorem subgroup.map_normalizer_eq_of_bijective {G : Type u_1} [group G] {N : Type u_3} [group N] (H : subgroup G) {f : G →* N} (hf : function.bijective f) :

The image of the normalizer is equal to the normalizer of the image of a bijective function.

The image of the normalizer is equal to the normalizer of the image of a bijective function.

def add_monoid_hom.lift_of_right_inverse_aux {G₁ : Type u_3} {G₂ : Type u_4} {G₃ : Type u_5} [add_group G₁] [add_group G₂] [add_group G₃] (f : G₁ →+ G₂) (f_inv : G₂ → G₁) (hf : function.right_inverse f_inv f) (g : G₁ →+ G₃) (hg : f.ker g.ker) :
G₂ →+ G₃

Auxiliary definition used to define lift_of_right_inverse

Equations
def monoid_hom.lift_of_right_inverse_aux {G₁ : Type u_3} {G₂ : Type u_4} {G₃ : Type u_5} [group G₁] [group G₂] [group G₃] (f : G₁ →* G₂) (f_inv : G₂ → G₁) (hf : function.right_inverse f_inv f) (g : G₁ →* G₃) (hg : f.ker g.ker) :
G₂ →* G₃

Auxiliary definition used to define lift_of_right_inverse

Equations
@[simp]
theorem monoid_hom.lift_of_right_inverse_aux_comp_apply {G₁ : Type u_3} {G₂ : Type u_4} {G₃ : Type u_5} [group G₁] [group G₂] [group G₃] (f : G₁ →* G₂) (f_inv : G₂ → G₁) (hf : function.right_inverse f_inv f) (g : G₁ →* G₃) (hg : f.ker g.ker) (x : G₁) :
(f.lift_of_right_inverse_aux f_inv hf g hg) (f x) = g x
@[simp]
theorem add_monoid_hom.lift_of_right_inverse_aux_comp_apply {G₁ : Type u_3} {G₂ : Type u_4} {G₃ : Type u_5} [add_group G₁] [add_group G₂] [add_group G₃] (f : G₁ →+ G₂) (f_inv : G₂ → G₁) (hf : function.right_inverse f_inv f) (g : G₁ →+ G₃) (hg : f.ker g.ker) (x : G₁) :
(f.lift_of_right_inverse_aux f_inv hf g hg) (f x) = g x
def add_monoid_hom.lift_of_right_inverse {G₁ : Type u_3} {G₂ : Type u_4} {G₃ : Type u_5} [add_group G₁] [add_group G₂] [add_group G₃] (f : G₁ →+ G₂) (f_inv : G₂ → G₁) (hf : function.right_inverse f_inv f) :
{g // f.ker g.ker} (G₂ →+ G₃)

lift_of_right_inverse f f_inv hf g hg is the unique additive group homomorphism φ

See add_monoid_hom.eq_lift_of_right_inverse for the uniqueness lemma.

   G₁.
   |  \
 f |   \ g
   |    \
   v     \⌟
   G₂----> G₃
      ∃!φ
Equations
def monoid_hom.lift_of_right_inverse {G₁ : Type u_3} {G₂ : Type u_4} {G₃ : Type u_5} [group G₁] [group G₂] [group G₃] (f : G₁ →* G₂) (f_inv : G₂ → G₁) (hf : function.right_inverse f_inv f) :
{g // f.ker g.ker} (G₂ →* G₃)

lift_of_right_inverse f hf g hg is the unique group homomorphism φ

See monoid_hom.eq_lift_of_right_inverse for the uniqueness lemma.

   G₁.
   |  \
 f |   \ g
   |    \
   v     \⌟
   G₂----> G₃
      ∃!φ
Equations
@[simp]
noncomputable def add_monoid_hom.lift_of_surjective {G₁ : Type u_3} {G₂ : Type u_4} {G₃ : Type u_5} [add_group G₁] [add_group G₂] [add_group G₃] (f : G₁ →+ G₂) (hf : function.surjective f) :
{g // f.ker g.ker} (G₂ →+ G₃)

A non-computable version of add_monoid_hom.lift_of_right_inverse for when no computable right inverse is available.

@[simp]
noncomputable def monoid_hom.lift_of_surjective {G₁ : Type u_3} {G₂ : Type u_4} {G₃ : Type u_5} [group G₁] [group G₂] [group G₃] (f : G₁ →* G₂) (hf : function.surjective f) :
{g // f.ker g.ker} (G₂ →* G₃)

A non-computable version of monoid_hom.lift_of_right_inverse for when no computable right inverse is available, that uses function.surj_inv.

@[simp]
theorem monoid_hom.lift_of_right_inverse_comp_apply {G₁ : Type u_3} {G₂ : Type u_4} {G₃ : Type u_5} [group G₁] [group G₂] [group G₃] (f : G₁ →* G₂) (f_inv : G₂ → G₁) (hf : function.right_inverse f_inv f) (g : {g // f.ker g.ker}) (x : G₁) :
((f.lift_of_right_inverse f_inv hf) g) (f x) = g x
@[simp]
theorem add_monoid_hom.lift_of_right_inverse_comp_apply {G₁ : Type u_3} {G₂ : Type u_4} {G₃ : Type u_5} [add_group G₁] [add_group G₂] [add_group G₃] (f : G₁ →+ G₂) (f_inv : G₂ → G₁) (hf : function.right_inverse f_inv f) (g : {g // f.ker g.ker}) (x : G₁) :
((f.lift_of_right_inverse f_inv hf) g) (f x) = g x
@[simp]
theorem add_monoid_hom.lift_of_right_inverse_comp {G₁ : Type u_3} {G₂ : Type u_4} {G₃ : Type u_5} [add_group G₁] [add_group G₂] [add_group G₃] (f : G₁ →+ G₂) (f_inv : G₂ → G₁) (hf : function.right_inverse f_inv f) (g : {g // f.ker g.ker}) :
((f.lift_of_right_inverse f_inv hf) g).comp f = g
@[simp]
theorem monoid_hom.lift_of_right_inverse_comp {G₁ : Type u_3} {G₂ : Type u_4} {G₃ : Type u_5} [group G₁] [group G₂] [group G₃] (f : G₁ →* G₂) (f_inv : G₂ → G₁) (hf : function.right_inverse f_inv f) (g : {g // f.ker g.ker}) :
((f.lift_of_right_inverse f_inv hf) g).comp f = g
theorem add_monoid_hom.eq_lift_of_right_inverse {G₁ : Type u_3} {G₂ : Type u_4} {G₃ : Type u_5} [add_group G₁] [add_group G₂] [add_group G₃] (f : G₁ →+ G₂) (f_inv : G₂ → G₁) (hf : function.right_inverse f_inv f) (g : G₁ →+ G₃) (hg : f.ker g.ker) (h : G₂ →+ G₃) (hh : h.comp f = g) :
h = (f.lift_of_right_inverse f_inv hf) g, hg⟩
theorem monoid_hom.eq_lift_of_right_inverse {G₁ : Type u_3} {G₂ : Type u_4} {G₃ : Type u_5} [group G₁] [group G₂] [group G₃] (f : G₁ →* G₂) (f_inv : G₂ → G₁) (hf : function.right_inverse f_inv f) (g : G₁ →* G₃) (hg : f.ker g.ker) (h : G₂ →* G₃) (hh : h.comp f = g) :
h = (f.lift_of_right_inverse f_inv hf) g, hg⟩
theorem subgroup.normal.comap {G : Type u_1} [group G] {N : Type u_3} [group N] {H : subgroup N} (hH : H.normal) (f : G →* N) :
theorem add_subgroup.normal.comap {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {H : add_subgroup N} (hH : H.normal) (f : G →+ N) :
@[protected, instance]
def subgroup.normal_comap {G : Type u_1} [group G] {N : Type u_3} [group N] {H : subgroup N} [nH : H.normal] (f : G →* N) :
@[protected, instance]
def add_subgroup.normal_comap {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {H : add_subgroup N} [nH : H.normal] (f : G →+ N) :
@[protected, instance]
def monoid_hom.normal_ker {G : Type u_1} [group G] {N : Type u_3} [group N] (f : G →* N) :
@[protected, instance]
def add_monoid_hom.normal_ker {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (f : G →+ N) :
@[protected, instance]
def subgroup.normal_inf {G : Type u_1} [group G] (H N : subgroup G) [hN : N.normal] :
@[protected, instance]
def add_subgroup.normal_inf {G : Type u_1} [add_group G] (H N : add_subgroup G) [hN : N.normal] :
def subgroup.zpowers {G : Type u_1} [group G] (g : G) :

The subgroup generated by an element.

Equations
@[simp]
theorem subgroup.mem_zpowers {G : Type u_1} [group G] (g : G) :
theorem subgroup.zpowers_eq_closure {G : Type u_1} [group G] (g : G) :
@[simp]
theorem subgroup.range_zpowers_hom {G : Type u_1} [group G] (g : G) :
theorem subgroup.zpowers_subset {G : Type u_1} [group G] {a : G} {K : subgroup G} (h : a K) :
theorem subgroup.mem_zpowers_iff {G : Type u_1} [group G] {g h : G} :
h subgroup.zpowers g ∃ (k : ), g ^ k = h
@[simp]
theorem subgroup.forall_zpowers {G : Type u_1} [group G] {x : G} {p : (subgroup.zpowers x) → Prop} :
(∀ (g : (subgroup.zpowers x)), p g) ∀ (m : ), p x ^ m, _⟩
@[simp]
theorem subgroup.exists_zpowers {G : Type u_1} [group G] {x : G} {p : (subgroup.zpowers x) → Prop} :
(∃ (g : (subgroup.zpowers x)), p g) ∃ (m : ), p x ^ m, _⟩
theorem subgroup.forall_mem_zpowers {G : Type u_1} [group G] {x : G} {p : G → Prop} :
(∀ (g : G), g subgroup.zpowers xp g) ∀ (m : ), p (x ^ m)
theorem subgroup.exists_mem_zpowers {G : Type u_1} [group G] {x : G} {p : G → Prop} :
(∃ (g : G) (H : g subgroup.zpowers x), p g) ∃ (m : ), p (x ^ m)
def add_subgroup.zmultiples {A : Type u_2} [add_group A] (a : A) :

The subgroup generated by an element.

Equations
@[simp]
@[simp]
theorem add_subgroup.mem_zmultiples {G : Type u_1} [add_group G] (g : G) :
theorem add_subgroup.zmultiples_subset {G : Type u_1} [add_group G] {a : G} {K : add_subgroup G} (h : a K) :
theorem add_subgroup.mem_zmultiples_iff {G : Type u_1} [add_group G] {g h : G} :
h add_subgroup.zmultiples g ∃ (k : ), k g = h
@[simp]
theorem add_subgroup.forall_zmultiples {G : Type u_1} [add_group G] {x : G} {p : (add_subgroup.zmultiples x) → Prop} :
(∀ (g : (add_subgroup.zmultiples x)), p g) ∀ (m : ), p m x, _⟩
theorem add_subgroup.forall_mem_zmultiples {G : Type u_1} [add_group G] {x : G} {p : G → Prop} :
(∀ (g : G), g add_subgroup.zmultiples xp g) ∀ (m : ), p (m x)
@[simp]
theorem add_subgroup.exists_zmultiples {G : Type u_1} [add_group G] {x : G} {p : (add_subgroup.zmultiples x) → Prop} :
(∃ (g : (add_subgroup.zmultiples x)), p g) ∃ (m : ), p m x, _⟩
theorem add_subgroup.exists_mem_zmultiples {G : Type u_1} [add_group G] {x : G} {p : G → Prop} :
(∃ (g : G) (H : g add_subgroup.zmultiples x), p g) ∃ (m : ), p (m x)
@[simp]
theorem add_monoid_hom.add_subgroup_comap_apply_coe {G : Type u_1} [add_group G] {G' : Type u_4} [add_group G'] (f : G →+ G') (H' : add_subgroup G') (x : (add_submonoid.comap f H'.to_add_submonoid)) :
def monoid_hom.subgroup_comap {G : Type u_1} [group G] {G' : Type u_4} [group G'] (f : G →* G') (H' : subgroup G') :

The monoid_hom from the preimage of a subgroup to itself.

Equations
def add_monoid_hom.add_subgroup_comap {G : Type u_1} [add_group G] {G' : Type u_4} [add_group G'] (f : G →+ G') (H' : add_subgroup G') :

the add_monoid_hom from the preimage of an additive subgroup to itself.

Equations
@[simp]
theorem monoid_hom.subgroup_comap_apply_coe {G : Type u_1} [group G] {G' : Type u_4} [group G'] (f : G →* G') (H' : subgroup G') (x : (submonoid.comap f H'.to_submonoid)) :
def monoid_hom.subgroup_map {G : Type u_1} [group G] {G' : Type u_4} [group G'] (f : G →* G') (H : subgroup G) :

The monoid_hom from a subgroup to its image.

Equations
@[simp]
theorem monoid_hom.subgroup_map_apply_coe {G : Type u_1} [group G] {G' : Type u_4} [group G'] (f : G →* G') (H : subgroup G) (x : (H.to_submonoid)) :
@[simp]
theorem add_monoid_hom.add_subgroup_map_apply_coe {G : Type u_1} [add_group G] {G' : Type u_4} [add_group G'] (f : G →+ G') (H : add_subgroup G) (x : (H.to_add_submonoid)) :
def add_monoid_hom.add_subgroup_map {G : Type u_1} [add_group G] {G' : Type u_4} [add_group G'] (f : G →+ G') (H : add_subgroup G) :

the add_monoid_hom from an additive subgroup to its image

Equations
theorem add_monoid_hom.add_subgroup_map_surjective {G : Type u_1} [add_group G] {G' : Type u_4} [add_group G'] (f : G →+ G') (H : add_subgroup G) :
theorem monoid_hom.subgroup_map_surjective {G : Type u_1} [group G] {G' : Type u_4} [group G'] (f : G →* G') (H : subgroup G) :
def mul_equiv.subgroup_congr {G : Type u_1} [group G] {H K : subgroup G} (h : H = K) :

Makes the identity isomorphism from a proof two subgroups of a multiplicative group are equal.

Equations
def add_equiv.add_subgroup_congr {G : Type u_1} [add_group G] {H K : add_subgroup G} (h : H = K) :

Makes the identity additive isomorphism from a proof two subgroups of an additive group are equal.

Equations
def add_equiv.add_subgroup_equiv_map {G : Type u_1} [add_group G] {G' : Type u_2} [add_group G'] (e : G ≃+ G') (H : add_subgroup G) :

An add_equiv φ between two additive groups G and G' induces an add_equiv between a subgroup H ≤ G and the subgroup φ(H) ≤ G'.

Equations
def mul_equiv.subgroup_equiv_map {G : Type u_1} [group G] {G' : Type u_2} [group G'] (e : G ≃* G') (H : subgroup G) :

A mul_equiv φ between two groups G and G' induces a mul_equiv between a subgroup H ≤ G and the subgroup φ(H) ≤ G'.

Equations
theorem add_subgroup.mem_sup {C : Type u_4} [add_comm_group C] {s t : add_subgroup C} {x : C} :
x s t ∃ (y : C) (H : y s) (z : C) (H : z t), y + z = x
theorem subgroup.mem_sup {C : Type u_4} [comm_group C] {s t : subgroup C} {x : C} :
x s t ∃ (y : C) (H : y s) (z : C) (H : z t), y * z = x
theorem add_subgroup.mem_sup' {C : Type u_4} [add_comm_group C] {s t : add_subgroup C} {x : C} :
x s t ∃ (y : s) (z : t), y + z = x
theorem subgroup.mem_sup' {C : Type u_4} [comm_group C] {s t : subgroup C} {x : C} :
x s t ∃ (y : s) (z : t), (y) * z = x
@[protected, instance]
@[protected, instance]
@[instance]
def is_simple_group.to_nontrivial (G : Type u_1) [group G] [self : is_simple_group G] :
@[class]
structure is_simple_group (G : Type u_1) [group G] :
Prop

A group is simple when it has exactly two normal subgroups.

@[class]
structure is_simple_add_group (A : Type u_2) [add_group A] :
Prop

An add_group is simple when it has exactly two normal add_subgroups.

@[instance]
theorem subgroup.normal.eq_bot_or_eq_top {G : Type u_1} [group G] [is_simple_group G] {H : subgroup G} (Hn : H.normal) :
H = H =
theorem add_subgroup.normal.eq_bot_or_eq_top {G : Type u_1} [add_group G] [is_simple_add_group G] {H : add_subgroup G} (Hn : H.normal) :
H = H =
@[protected, instance]
theorem is_simple_group.is_simple_group_of_surjective {G : Type u_1} [group G] {H : Type u_2} [group H] [is_simple_group G] [nontrivial H] (f : G →* H) (hf : function.surjective f) :
theorem subgroup.sup_eq_closure {G : Type u_1} [group G] (H K : subgroup G) :
theorem add_subgroup.sup_eq_closure {G : Type u_1} [add_group G] (H K : add_subgroup G) :
theorem add_subgroup.add_normal {G : Type u_1} [add_group G] (H N : add_subgroup G) [N.normal] :
(H N) = H + N

The carrier of H ⊔ N is just ↑H + ↑N (pointwise set addition) when N is normal.

theorem subgroup.mul_normal {G : Type u_1} [group G] (H N : subgroup G) [N.normal] :
(H N) = (H) * N

The carrier of H ⊔ N is just ↑H * ↑N (pointwise set product) when N is normal.

theorem add_subgroup.normal_add {G : Type u_1} [add_group G] (N H : add_subgroup G) [N.normal] :
(N H) = N + H

The carrier of N ⊔ H is just ↑N + ↑H (pointwise set addition) when N is normal.

theorem subgroup.normal_mul {G : Type u_1} [group G] (N H : subgroup G) [N.normal] :
(N H) = (N) * H

The carrier of N ⊔ H is just ↑N * ↑H (pointwise set product) when N is normal.

theorem subgroup.mul_inf_assoc {G : Type u_1} [group G] (A B C : subgroup G) (h : A C) :
(A) * (B C) = (A) * B C
theorem add_subgroup.add_inf_assoc {G : Type u_1} [add_group G] (A B C : add_subgroup G) (h : A C) :
A + (B C) = (A + B) C
theorem subgroup.inf_mul_assoc {G : Type u_1} [group G] (A B C : subgroup G) (h : C A) :
((A B)) * C = A (B) * C
theorem add_subgroup.inf_add_assoc {G : Type u_1} [add_group G] (A B C : add_subgroup G) (h : C A) :
(A B) + C = A (B + C)
theorem subgroup.normal_subgroup_of_iff {G : Type u_1} [group G] {H K : subgroup G} (hHK : H K) :
(H.subgroup_of K).normal ∀ (h k : G), h Hk K(k * h) * k⁻¹ H
theorem add_subgroup.normal_add_subgroup_of_iff {G : Type u_1} [add_group G] {H K : add_subgroup G} (hHK : H K) :
(H.add_subgroup_of K).normal ∀ (h k : G), h Hk Kk + h + -k H
@[protected, instance]
def add_subgroup.sum_add_subgroup_of_sum_normal {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] {H₁ K₁ : add_subgroup G} {H₂ K₂ : add_subgroup N} [h₁ : (H₁.add_subgroup_of K₁).normal] [h₂ : (H₂.add_subgroup_of K₂).normal] :
((H₁.prod H₂).add_subgroup_of (K₁.prod K₂)).normal
@[protected, instance]
def subgroup.prod_subgroup_of_prod_normal {G : Type u_1} [group G] {N : Type u_3} [group N] {H₁ K₁ : subgroup G} {H₂ K₂ : subgroup N} [h₁ : (H₁.subgroup_of K₁).normal] [h₂ : (H₂.subgroup_of K₂).normal] :
((H₁.prod H₂).subgroup_of (K₁.prod K₂)).normal
@[protected, instance]
def add_subgroup.sum_normal {G : Type u_1} [add_group G] {N : Type u_3} [add_group N] (H : add_subgroup G) (K : add_subgroup N) [hH : H.normal] [hK : K.normal] :
(H.prod K).normal
@[protected, instance]
def subgroup.prod_normal {G : Type u_1} [group G] {N : Type u_3} [group N] (H : subgroup G) (K : subgroup N) [hH : H.normal] [hK : K.normal] :
(H.prod K).normal
theorem subgroup.inf_subgroup_of_inf_normal_of_right {G : Type u_1} [group G] (A B' B : subgroup G) (hB : B' B) [hN : (B'.subgroup_of B).normal] :
((A B').subgroup_of (A B)).normal
theorem add_subgroup.inf_add_subgroup_of_inf_normal_of_right {G : Type u_1} [add_group G] (A B' B : add_subgroup G) (hB : B' B) [hN : (B'.add_subgroup_of B).normal] :
((A B').add_subgroup_of (A B)).normal
theorem subgroup.inf_subgroup_of_inf_normal_of_left {G : Type u_1} [group G] {A' A : subgroup G} (B : subgroup G) (hA : A' A) [hN : (A'.subgroup_of A).normal] :
((A' B).subgroup_of (A B)).normal
theorem add_subgroup.inf_add_subgroup_of_inf_normal_of_left {G : Type u_1} [add_group G] {A' A : add_subgroup G} (B : add_subgroup G) (hA : A' A) [hN : (A'.add_subgroup_of A).normal] :
((A' B).add_subgroup_of (A B)).normal
@[protected, instance]
def subgroup.sup_normal {G : Type u_1} [group G] (H K : subgroup G) [hH : H.normal] [hK : K.normal] :
(H K).normal
@[protected, instance]
def subgroup.normal_inf_normal {G : Type u_1} [group G] (H K : subgroup G) [hH : H.normal] [hK : K.normal] :
(H K).normal
@[protected, instance]
def add_subgroup.normal_inf_normal {G : Type u_1} [add_group G] (H K : add_subgroup G) [hH : H.normal] [hK : K.normal] :
(H K).normal
theorem add_subgroup.add_subgroup_of_sup {G : Type u_1} [add_group G] (A A' B : add_subgroup G) (hA : A B) (hA' : A' B) :
theorem subgroup.subgroup_of_sup {G : Type u_1} [group G] (A A' B : subgroup G) (hA : A B) (hA' : A' B) :
theorem add_subgroup.subgroup_normal.mem_comm {G : Type u_1} [add_group G] {H K : add_subgroup G} (hK : H K) [hN : (H.add_subgroup_of K).normal] {a b : G} (hb : b K) (h : a + b H) :
b + a H
theorem subgroup.subgroup_normal.mem_comm {G : Type u_1} [group G] {H K : subgroup G} (hK : H K) [hN : (H.subgroup_of K).normal] {a b : G} (hb : b K) (h : a * b H) :
b * a H
theorem is_conj.normal_closure_eq_top_of {G : Type u_1} [group G] {N : subgroup G} [hn : N.normal] {g g' : G} {hg : g N} {hg' : g' N} (hc : is_conj g g') (ht : subgroup.normal_closure {g, hg⟩} = ) :

Actions by subgroups #

These are just copies of the definitions about submonoid starting from submonoid.mul_action.

@[protected, instance]
def add_subgroup.add_action {G : Type u_1} [add_group G] {α : Type u_4} [add_action G α] (S : add_subgroup G) :

The additive action by an add_subgroup is the action by the underlying add_group.

Equations
@[protected, instance]
def subgroup.mul_action {G : Type u_1} [group G] {α : Type u_4} [mul_action G α] (S : subgroup G) :

The action by a subgroup is the action by the underlying group.

Equations
theorem subgroup.smul_def {G : Type u_1} [group G] {α : Type u_4} [mul_action G α] {S : subgroup G} (g : S) (m : α) :
g m = g m
theorem add_subgroup.vadd_def {G : Type u_1} [add_group G] {α : Type u_4} [add_action G α] {S : add_subgroup G} (g : S) (m : α) :
g +ᵥ m = g +ᵥ m
@[protected, instance]
def subgroup.smul_comm_class_left {G : Type u_1} [group G] {α : Type u_4} {β : Type u_5} [mul_action G β] [has_scalar α β] [smul_comm_class G α β] (S : subgroup G) :
@[protected, instance]
def add_subgroup.vadd_comm_class_left {G : Type u_1} [add_group G] {α : Type u_4} {β : Type u_5} [add_action G β] [has_vadd α β] [vadd_comm_class G α β] (S : add_subgroup G) :
@[protected, instance]
def subgroup.smul_comm_class_right {G : Type u_1} [group G] {α : Type u_4} {β : Type u_5} [has_scalar α β] [mul_action G β] [smul_comm_class α G β] (S : subgroup G) :
@[protected, instance]
def add_subgroup.vadd_comm_class_right {G : Type u_1} [add_group G] {α : Type u_4} {β : Type u_5} [has_vadd α β] [add_action G β] [vadd_comm_class α G β] (S : add_subgroup G) :
@[protected, instance]
def subgroup.is_scalar_tower {G : Type u_1} [group G] {α : Type u_4} {β : Type u_5} [has_scalar α β] [mul_action G α] [mul_action G β] [is_scalar_tower G α β] (S : subgroup G) :

Note that this provides is_scalar_tower S G G which is needed by smul_mul_assoc.

@[protected, instance]
def subgroup.has_faithful_scalar {G : Type u_1} [group G] {α : Type u_4} [mul_action G α] [has_faithful_scalar G α] (S : subgroup G) :
@[protected, instance]
def subgroup.distrib_mul_action {G : Type u_1} [group G] {α : Type u_4} [add_monoid α] [distrib_mul_action G α] (S : subgroup G) :

The action by a subgroup is the action by the underlying group.

Equations
@[protected, instance]
def subgroup.mul_distrib_mul_action {G : Type u_1} [group G] {α : Type u_4} [monoid α] [mul_distrib_mul_action G α] (S : subgroup G) :

The action by a subgroup is the action by the underlying group.

Equations

Saturated subgroups #

def add_subgroup.saturated {G : Type u_1} [add_group G] (H : add_subgroup G) :
Prop

An additive subgroup H of G is saturated if for all n : ℕ and g : G with n•g ∈ H we have n = 0 or g ∈ H.

Equations
def subgroup.saturated {G : Type u_1} [group G] (H : subgroup G) :
Prop

A subgroup H of G is saturated if for all n : ℕ and g : G with g^n ∈ H we have n = 0 or g ∈ H.

Equations
theorem subgroup.saturated_iff_npow {G : Type u_1} [group G] {H : subgroup G} :
H.saturated ∀ (n : ) (g : G), g ^ n Hn = 0 g H
theorem add_subgroup.saturated_iff_nsmul {G : Type u_1} [add_group G] {H : add_subgroup G} :
H.saturated ∀ (n : ) (g : G), n g Hn = 0 g H
theorem add_subgroup.saturated_iff_zsmul {G : Type u_1} [add_group G] {H : add_subgroup G} :
H.saturated ∀ (n : ) (g : G), n g Hn = 0 g H
theorem subgroup.saturated_iff_zpow {G : Type u_1} [group G] {H : subgroup G} :
H.saturated ∀ (n : ) (g : G), g ^ n Hn = 0 g H
theorem add_subgroup.ker_saturated {A₁ : Type u_1} {A₂ : Type u_2} [add_comm_group A₁] [add_comm_group A₂] [no_zero_smul_divisors A₂] (f : A₁ →+ A₂) :