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:
-
H Karesubgroups ofGoradd_subgroups ofA -
s kare sets of elements of typeG
Definitions in the file:
-
add_subgroup A: the type of subgroups of an additive groupA -
complete_lattice (subgroup G): the subgroups ofGform a complete lattice -
subgroup.closure k: the minimal subgroup that includes the setk -
subgroup.subtype: the natural group homomorphism from a subgroup of groupGtoG -
subgroup.gi:closureforms a Galois insertion with the coercion to set -
subgroup.comap H f: the preimage of a subgroupHalong the group homomorphismfis also a subgroup -
subgroup.map f H: the image of a subgroupHalong the group homomorphismfis also a subgroup -
subgroup.prod H K: the product of subgroupsH,Kof groupsG,Nrespectively,H × Kis a subgroup ofG × N -
monoid_hom.range f: the range of the group homomorphismfis a subgroup -
monoid_hom.ker f: the kernel of a group homomorphismfis the subgroup of elementsx : Gsuch thatf x = 1 -
monoid_hom.eq_locus f g: given group homomorphismsf,g, the elements ofGsuch thatf x = g xform a subgroup ofG -
is_simple_group G: a class indicating that a group has exactly two normal subgroups
Implementation notes #
Subgroup inclusion is denoted ≤ rather than ⊆, although ∈ is defined as
membership of a subgroup's underlying set.
Tags #
subgroup, subgroups
- carrier : set G
- one_mem' : 1 ∈ self.carrier
- mul_mem' : ∀ {a b : G}, a ∈ self.carrier → b ∈ self.carrier → a * b ∈ self.carrier
- inv_mem' : ∀ {x : G}, x ∈ self.carrier → x⁻¹ ∈ self.carrier
A subgroup of a group G is a subset containing 1, closed under multiplication
and closed under multiplicative inverse.
- carrier : set G
- zero_mem' : 0 ∈ self.carrier
- add_mem' : ∀ {a b : G}, a ∈ self.carrier → b ∈ self.carrier → a + b ∈ self.carrier
- neg_mem' : ∀ {x : G}, x ∈ self.carrier → -x ∈ self.carrier
An additive subgroup of an additive group G is a subset containing 0, closed
under addition and additive inverse.
Reinterpret an add_subgroup as an add_submonoid.
@[protected, instance]
Equations
- subgroup.set_like = {coe := subgroup.carrier _inst_1, coe_injective' := _}
@[protected, instance]
Equations
- add_subgroup.set_like = {coe := add_subgroup.carrier _inst_1, coe_injective' := _}
@[simp]
See Note [custom simps projection]
Equations
- subgroup.simps.coe S = ↑S
See Note [custom simps projection]
Equations
@[simp]
theorem
add_subgroup.coe_to_add_submonoid
{G : Type u_1}
[add_group G]
(K : add_subgroup G) :
↑(K.to_add_submonoid) = ↑K
@[simp]
theorem
subgroup.coe_to_submonoid
{G : Type u_1}
[group G]
(K : subgroup G) :
↑(K.to_submonoid) = ↑K
@[simp]
theorem
subgroup.mem_to_submonoid
{G : Type u_1}
[group G]
(K : subgroup G)
(x : G) :
x ∈ K.to_submonoid ↔ x ∈ K
@[simp]
theorem
add_subgroup.mem_to_add_submonoid
{G : Type u_1}
[add_group G]
(K : add_subgroup G)
(x : G) :
x ∈ K.to_add_submonoid ↔ x ∈ K
@[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
- K.fintype = show fintype {g // g ∈ K}, from infer_instance
@[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
- K.fintype = show fintype {g // g ∈ K}, from infer_instance
@[simp]
theorem
add_subgroup.to_add_submonoid_eq
{G : Type u_1}
[add_group G]
{p q : add_subgroup G} :
p.to_add_submonoid = q.to_add_submonoid ↔ p = q
@[simp]
theorem
subgroup.to_submonoid_eq
{G : Type u_1}
[group G]
{p q : subgroup G} :
p.to_submonoid = q.to_submonoid ↔ p = q
@[simp]
theorem
add_subgroup.to_add_submonoid_le
{G : Type u_1}
[add_group G]
{p q : add_subgroup G} :
p.to_add_submonoid ≤ q.to_add_submonoid ↔ p ≤ q
@[simp]
theorem
subgroup.to_submonoid_le
{G : Type u_1}
[group G]
{p q : subgroup G} :
p.to_submonoid ≤ q.to_submonoid ↔ p ≤ q
Conversion to/from additive/multiplicative #
Supgroups of a group G are isomorphic to additive subgroups of additive G.
Equations
- subgroup.to_add_subgroup = {to_equiv := {to_fun := λ (S : subgroup G), {carrier := (⇑submonoid.to_add_submonoid S.to_submonoid).carrier, zero_mem' := _, add_mem' := _, neg_mem' := _}, inv_fun := λ (S : add_subgroup (additive G)), {carrier := (⇑add_submonoid.to_submonoid' S.to_add_submonoid).carrier, one_mem' := _, mul_mem' := _, inv_mem' := _}, left_inv := _, right_inv := _}, map_rel_iff' := _}
@[simp]
@[simp]
theorem
subgroup.to_add_subgroup_symm_apply_coe
{G : Type u_1}
[group G]
(S : add_subgroup (additive G)) :
def
add_subgroup.to_subgroup
{A : Type u_2}
[add_group A] :
add_subgroup A ≃o subgroup (multiplicative A)
Additive supgroups of an additive group A are isomorphic to subgroups of multiplicative A.
Equations
- add_subgroup.to_subgroup = {to_equiv := {to_fun := λ (S : add_subgroup A), {carrier := (⇑add_submonoid.to_submonoid S.to_add_submonoid).carrier, one_mem' := _, mul_mem' := _, inv_mem' := _}, inv_fun := λ (S : subgroup (multiplicative A)), {carrier := (⇑submonoid.to_add_submonoid' S.to_submonoid).carrier, zero_mem' := _, add_mem' := _, neg_mem' := _}, left_inv := _, right_inv := _}, map_rel_iff' := _}
@[simp]
@[simp]
theorem
add_subgroup.to_subgroup_symm_apply_coe
{A : Type u_2}
[add_group A]
(S : subgroup (multiplicative A)) :
def
subgroup.to_add_subgroup'
{A : Type u_2}
[add_group A] :
subgroup (multiplicative A) ≃o add_subgroup A
Subgroups of an additive group multiplicative A are isomorphic to additive subgroups of A.
@[protected]
Copy of a subgroup with a new carrier equal to the old one. Useful to fix definitional
equalities.
@[protected]
Copy of an additive subgroup with a new carrier equal to the old one.
Useful to fix definitional equalities
@[simp]
theorem
add_subgroup.coe_copy
{G : Type u_1}
[add_group G]
(K : add_subgroup G)
(s : set G)
(hs : s = ↑K) :
theorem
add_subgroup.copy_eq
{G : Type u_1}
[add_group G]
(K : add_subgroup G)
(s : set G)
(hs : s = ↑K) :
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.
A subgroup contains the group's 1.
An add_subgroup contains the group's 0.
An add_subgroup is closed under addition.
An add_subgroup is closed under inverse.
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) :
An add_subgroup is closed under subtraction.
@[simp]
@[simp]
@[simp]
theorem
add_subgroup.exists_neg_mem_iff_exists_mem
{G : Type u_1}
[add_group G]
(K : add_subgroup G)
{P : G → Prop} :
theorem
add_subgroup.add_mem_cancel_right
{G : Type u_1}
[add_group G]
(H : add_subgroup G)
{x y : G}
(h : x ∈ 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) :
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) :
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) :
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 ∈ t → f 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 ∈ t → f 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
add_subgroup.nsmul_mem
{G : Type u_1}
[add_group G]
(K : add_subgroup G)
{x : G}
(hx : x ∈ K)
(n : ℕ) :
theorem
add_subgroup.zsmul_mem
{G : Type u_1}
[add_group G]
(K : add_subgroup G)
{x : G}
(hx : x ∈ K)
(n : ℤ) :
@[protected, instance]
A subgroup of a group inherits a multiplication.
Equations
- H.has_mul = H.to_submonoid.has_mul
@[protected, instance]
A subgroup of a group inherits a 1.
Equations
- H.has_one = H.to_submonoid.has_one
@[simp, norm_cast]
@[simp, norm_cast]
@[simp, norm_cast]
theorem
add_subgroup.coe_mk
{G : Type u_1}
[add_group G]
(H : add_subgroup G)
(x : G)
(hx : x ∈ H) :
@[protected, instance]
A subgroup of a group inherits a group structure.
Equations
- H.to_group = function.injective.group coe _ _ _ _ _
@[protected, instance]
An add_subgroup of an add_group inherits an add_group structure.
Equations
- H.to_add_group = function.injective.add_group coe _ _ _ _ _
@[protected, instance]
An add_subgroup of an add_comm_group is an add_comm_group.
Equations
- H.to_add_comm_group = function.injective.add_comm_group coe _ _ _ _ _
@[protected, instance]
A subgroup of a comm_group is a comm_group.
Equations
- H.to_comm_group = function.injective.comm_group coe _ _ _ _ _
@[protected, instance]
def
add_subgroup.to_ordered_add_comm_group
{G : Type u_1}
[ordered_add_comm_group G]
(H : add_subgroup G) :
An add_subgroup of an add_ordered_comm_group is an add_ordered_comm_group.
Equations
@[protected, instance]
A subgroup of an ordered_comm_group is an ordered_comm_group.
Equations
@[protected, instance]
def
subgroup.to_linear_ordered_comm_group
{G : Type u_1}
[linear_ordered_comm_group G]
(H : subgroup G) :
A subgroup of a linear_ordered_comm_group is a linear_ordered_comm_group.
Equations
@[protected, instance]
def
add_subgroup.to_linear_ordered_add_comm_group
{G : Type u_1}
[linear_ordered_add_comm_group G]
(H : add_subgroup G) :
An add_subgroup of a linear_ordered_add_comm_group is a
linear_ordered_add_comm_group.
Equations
The natural group hom from an add_subgroup of add_group G to G.
@[simp]
@[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) :
@[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) :
The inclusion homomorphism from a additive subgroup H contained in K to K.
Equations
- add_subgroup.inclusion h = add_monoid_hom.mk' (λ (x : ↥H), ⟨↑x, _⟩) _
The inclusion homomorphism from a subgroup H contained in K to K.
Equations
- subgroup.inclusion h = monoid_hom.mk' (λ (x : ↥H), ⟨↑x, _⟩) _
@[simp]
theorem
add_subgroup.coe_inclusion
{G : Type u_1}
[add_group G]
{H K : add_subgroup G}
{h : H ≤ K}
(a : ↥H) :
↑(⇑(add_subgroup.inclusion h) a) = ↑a
@[simp]
theorem
add_subgroup.subtype_comp_inclusion
{G : Type u_1}
[add_group G]
{H K : add_subgroup G}
(hH : H ≤ K) :
K.subtype.comp (add_subgroup.inclusion hH) = H.subtype
@[simp]
theorem
subgroup.subtype_comp_inclusion
{G : Type u_1}
[group G]
{H K : subgroup G}
(hH : H ≤ K) :
K.subtype.comp (subgroup.inclusion hH) = H.subtype
@[protected, instance]
The add_subgroup G of the add_group G.
@[protected, instance]
The trivial add_subgroup {0} of an add_group G.
@[protected, instance]
Equations
@[protected, instance]
Equations
- subgroup.inhabited = {default := ⊥}
@[protected, instance]
Equations
- add_subgroup.has_bot.bot.unique = {to_inhabited := {default := 0}, uniq := _}
@[protected, instance]
Equations
- subgroup.has_bot.bot.unique = {to_inhabited := {default := 1}, uniq := _}
theorem
subgroup.eq_bot_of_subsingleton
{G : Type u_1}
[group G]
(H : subgroup G)
[subsingleton ↥H] :
theorem
add_subgroup.eq_bot_of_subsingleton
{G : Type u_1}
[add_group G]
(H : add_subgroup G)
[subsingleton ↥H] :
@[simp]
@[simp]
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) :
theorem
add_subgroup.eq_top_of_card_eq
{G : Type u_1}
[add_group G]
(H : add_subgroup G)
[fintype ↥H]
[fintype G]
(h : fintype.card ↥H = fintype.card G) :
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) :
theorem
add_subgroup.eq_top_of_le_card
{G : Type u_1}
[add_group G]
(H : add_subgroup G)
[fintype ↥H]
[fintype G]
(h : fintype.card G ≤ fintype.card ↥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) :
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) :
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) :
H = ⊥ ∨ nontrivial ↥H
A subgroup is either the trivial subgroup or nontrivial.
theorem
add_subgroup.bot_or_nontrivial
{G : Type u_1}
[add_group G]
(H : add_subgroup G) :
H = ⊥ ∨ nontrivial ↥H
theorem
add_subgroup.card_nonpos_iff_eq_bot
{G : Type u_1}
[add_group G]
(H : add_subgroup G)
[fintype ↥H] :
theorem
add_subgroup.pos_card_iff_ne_bot
{G : Type u_1}
[add_group G]
(H : add_subgroup G)
[fintype ↥H] :
@[protected, instance]
The inf of two subgroups is their intersection.
Equations
- subgroup.has_inf = {inf := λ (H₁ H₂ : subgroup G), {carrier := (H₁.to_submonoid ⊓ H₂.to_submonoid).carrier, one_mem' := _, mul_mem' := _, inv_mem' := _}}
@[protected, instance]
The inf of two add_subgroupss is their intersection.
Equations
- add_subgroup.has_inf = {inf := λ (H₁ H₂ : add_subgroup G), {carrier := (H₁.to_add_submonoid ⊓ H₂.to_add_submonoid).carrier, zero_mem' := _, add_mem' := _, neg_mem' := _}}
@[simp]
@[simp]
@[protected, instance]
Equations
- add_subgroup.has_Inf = {Inf := λ (s : set (add_subgroup G)), {carrier := ((⨅ (S : add_subgroup G) (H : S ∈ s), S.to_add_submonoid).copy (⋂ (S : add_subgroup G) (H : S ∈ s), ↑S) _).carrier, zero_mem' := _, add_mem' := _, neg_mem' := _}}
@[protected, instance]
@[simp, norm_cast]
@[simp]
theorem
add_subgroup.mem_infi
{G : Type u_1}
[add_group G]
{ι : Sort u_2}
{S : ι → add_subgroup G}
{x : G} :
@[simp, norm_cast]
theorem
add_subgroup.coe_infi
{G : Type u_1}
[add_group G]
{ι : Sort u_2}
{S : ι → add_subgroup G} :
@[protected, instance]
Subgroups of a group form a complete lattice.
Equations
- subgroup.complete_lattice = {sup := complete_lattice.sup (complete_lattice_of_Inf (subgroup G) subgroup.complete_lattice._proof_1), le := complete_lattice.le (complete_lattice_of_Inf (subgroup G) subgroup.complete_lattice._proof_1), lt := complete_lattice.lt (complete_lattice_of_Inf (subgroup G) subgroup.complete_lattice._proof_1), le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, le_sup_left := _, le_sup_right := _, sup_le := _, inf := has_inf.inf subgroup.has_inf, inf_le_left := _, inf_le_right := _, le_inf := _, Sup := complete_lattice.Sup (complete_lattice_of_Inf (subgroup G) subgroup.complete_lattice._proof_1), le_Sup := _, Sup_le := _, Inf := complete_lattice.Inf (complete_lattice_of_Inf (subgroup G) subgroup.complete_lattice._proof_1), Inf_le := _, le_Inf := _, top := ⊤, bot := ⊥, le_top := _, bot_le := _}
@[protected, instance]
The add_subgroups of an add_group form a complete lattice.
Equations
- add_subgroup.complete_lattice = {sup := complete_lattice.sup (complete_lattice_of_Inf (add_subgroup G) add_subgroup.complete_lattice._proof_1), le := complete_lattice.le (complete_lattice_of_Inf (add_subgroup G) add_subgroup.complete_lattice._proof_1), lt := complete_lattice.lt (complete_lattice_of_Inf (add_subgroup G) add_subgroup.complete_lattice._proof_1), le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, le_sup_left := _, le_sup_right := _, sup_le := _, inf := has_inf.inf add_subgroup.has_inf, inf_le_left := _, inf_le_right := _, le_inf := _, Sup := complete_lattice.Sup (complete_lattice_of_Inf (add_subgroup G) add_subgroup.complete_lattice._proof_1), le_Sup := _, Sup_le := _, Inf := complete_lattice.Inf (complete_lattice_of_Inf (add_subgroup G) add_subgroup.complete_lattice._proof_1), Inf_le := _, le_Inf := _, top := ⊤, bot := ⊥, le_top := _, bot_le := _}
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) :
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} :
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} :
@[simp]
theorem
subgroup.subsingleton_iff
{G : Type u_1}
[group G] :
subsingleton (subgroup G) ↔ subsingleton G
@[simp]
@[simp]
@[simp]
theorem
add_subgroup.nontrivial_iff
{G : Type u_1}
[add_group G] :
nontrivial (add_subgroup G) ↔ nontrivial G
@[protected, instance]
Equations
- add_subgroup.unique = {to_inhabited := {default := ⊥}, uniq := _}
@[protected, instance]
Equations
- subgroup.unique = {to_inhabited := {default := ⊥}, uniq := _}
@[protected, instance]
@[protected, instance]
The add_subgroup generated by a set
Equations
- add_subgroup.closure k = Inf {K : add_subgroup G | k ⊆ ↑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 ⊆ ↑K → x ∈ K
@[simp]
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} :
k ⊆ ↑(add_subgroup.closure k)
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} :
subgroup.closure k ≤ K ↔ k ⊆ ↑K
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} :
add_subgroup.closure k ≤ K ↔ k ⊆ ↑K
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) :
subgroup.closure k = 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 ∈ k → p x)
(H1 : p 1)
(Hmul : ∀ (x y : G), p x → p y → p (x * y))
(Hinv : ∀ (x : G), p x → p 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 ∈ k → p x)
(H1 : p 0)
(Hmul : ∀ (x y : G), p x → p y → p (x + y))
(Hinv : ∀ (x : G), p x → p (-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 x → p y → p (x * y))
(Hinv : ∀ (x : ↥(subgroup.closure k)), p x → p 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 x → p y → p (x + y))
(Hinv : ∀ (x : ↥(add_subgroup.closure k)), p x → p (-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} :
subgroup.closure (coe ⁻¹' k) = ⊤
@[simp]
theorem
add_subgroup.closure_closure_coe_preimage
{G : Type u_1}
[add_group G]
{k : set G} :
add_subgroup.closure (coe ⁻¹' k) = ⊤
@[protected]
closure forms a Galois insertion with the coercion to set.
Equations
- add_subgroup.gi G = {choice := λ (s : set G) (_x : ↑(add_subgroup.closure s) ≤ s), add_subgroup.closure s, gc := _, le_l_u := _, choice_eq := _}
@[protected]
closure forms a Galois insertion with the coercion to set.
Equations
- subgroup.gi G = {choice := λ (s : set G) (_x : ↑(subgroup.closure s) ≤ s), subgroup.closure s, gc := _, le_l_u := _, choice_eq := _}
Subgroup closure of a set is monotone in its argument: if h ⊆ k,
then closure h ≤ closure k.
Additive subgroup closure of a set is monotone in its argument: if h ⊆ k,
then closure h ≤ closure k
@[simp]
Closure of a subgroup K equals K.
@[simp]
Additive closure of an additive subgroup K equals K
@[simp]
@[simp]
@[simp]
theorem
subgroup.closure_union
{G : Type u_1}
[group G]
(s t : set G) :
subgroup.closure (s ∪ t) = subgroup.closure s ⊔ subgroup.closure t
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) :
subgroup.closure S = ⊥ ↔ S ⊆ {1}
theorem
add_subgroup.closure_eq_bot_iff
(G : Type u_1)
[add_group G]
(S : set G) :
add_subgroup.closure S = ⊥ ↔ S ⊆ {0}
The add_subgroup generated by an element of an add_group equals the set of
natural number multiples of the element.
The subgroup generated by an element of a group equals the set of integer number powers of the element.
theorem
add_subgroup.closure_singleton_zero
{G : Type u_1}
[add_group G] :
add_subgroup.closure {0} = ⊥
@[simp]
theorem
subgroup.inv_subset_closure
{G : Type u_1}
[group G]
(S : set G) :
S⁻¹ ⊆ ↑(subgroup.closure S)
@[simp]
theorem
add_subgroup.neg_subset_closure
{G : Type u_1}
[add_group G]
(S : set G) :
-S ⊆ ↑(add_subgroup.closure S)
@[simp]
@[simp]
theorem
subgroup.closure_to_submonoid
{G : Type u_1}
[group G]
(S : set G) :
(subgroup.closure S).to_submonoid = submonoid.closure (S ∪ S⁻¹)
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 ∈ k → p x)
(Hk_inv : ∀ (x : G), x ∈ k → p x⁻¹)
(H1 : p 1)
(Hmul : ∀ (x y : G), p x → p y → p (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 ∈ k → p x)
(Hk_inv : ∀ (x : G), x ∈ k → p (-x))
(H1 : p 0)
(Hmul : ∀ (x y : G), p x → p y → p (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.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) :
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} :
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.
@[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} :
x ∈ add_subgroup.comap f K ↔ ⇑f x ∈ K
theorem
subgroup.comap_mono
{G : Type u_1}
[group G]
{N : Type u_3}
[group N]
{f : G →* N}
{K K' : subgroup N} :
K ≤ K' → subgroup.comap f K ≤ subgroup.comap f K'
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} :
K ≤ K' → add_subgroup.comap f K ≤ add_subgroup.comap f K'
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) :
add_subgroup.comap f (add_subgroup.comap g K) = add_subgroup.comap (g.comp f) K
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) :
subgroup.comap f (subgroup.comap g K) = subgroup.comap (g.comp f) K
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.
@[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
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} :
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) :
⇑f x ∈ add_subgroup.map f 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) :
⇑f x ∈ subgroup.map f 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) :
⇑f ↑x ∈ add_subgroup.map f 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} :
K ≤ K' → add_subgroup.map f K ≤ add_subgroup.map f K'
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) :
add_subgroup.map (add_monoid_hom.id G) K = K
@[simp]
theorem
subgroup.map_id
{G : Type u_1}
[group G]
(K : subgroup G) :
subgroup.map (monoid_hom.id G) K = K
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) :
add_subgroup.map g (add_subgroup.map f K) = add_subgroup.map (g.comp f) K
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) :
subgroup.map g (subgroup.map f K) = subgroup.map (g.comp f) K
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} :
x ∈ subgroup.map f.to_monoid_hom K ↔ ⇑(f.symm) x ∈ K
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} :
x ∈ add_subgroup.map f.to_add_monoid_hom K ↔ ⇑(f.symm) x ∈ K
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} :
⇑f x ∈ subgroup.map f K ↔ x ∈ K
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} :
⇑f x ∈ add_subgroup.map f K ↔ x ∈ K
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
add_subgroup.map_equiv_eq_comap_symm
{G : Type u_1}
[add_group G]
{N : Type u_3}
[add_group N]
(f : G ≃+ N)
(K : add_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.comap_equiv_eq_map_symm
{G : Type u_1}
[add_group G]
{N : Type u_3}
[add_group N]
(f : N ≃+ G)
(K : add_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} :
add_subgroup.map f K ≤ H ↔ K ≤ add_subgroup.comap f H
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} :
subgroup.map f K ≤ H ↔ K ≤ subgroup.comap f H
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) :
add_subgroup.map f (H ⊔ K) = add_subgroup.map f H ⊔ add_subgroup.map f K
theorem
subgroup.map_sup
{G : Type u_1}
[group G]
{N : Type u_3}
[group N]
(H K : subgroup G)
(f : G →* N) :
subgroup.map f (H ⊔ K) = subgroup.map f H ⊔ subgroup.map f K
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) :
add_subgroup.comap f H ⊔ add_subgroup.comap f K ≤ add_subgroup.comap f (H ⊔ K)
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) :
subgroup.comap f H ⊔ subgroup.comap f K ≤ subgroup.comap f (H ⊔ K)
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) :
add_subgroup.comap f (H ⊓ K) = add_subgroup.comap f H ⊓ add_subgroup.comap f K
theorem
subgroup.comap_inf
{G : Type u_1}
[group G]
{N : Type u_3}
[group N]
(H K : subgroup N)
(f : G →* N) :
subgroup.comap f (H ⊓ K) = subgroup.comap f H ⊓ subgroup.comap f K
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) :
add_subgroup.map f (H ⊓ K) ≤ add_subgroup.map f H ⊓ add_subgroup.map f K
theorem
subgroup.map_inf_le
{G : Type u_1}
[group G]
{N : Type u_3}
[group N]
(H K : subgroup G)
(f : G →* N) :
subgroup.map f (H ⊓ K) ≤ subgroup.map f H ⊓ subgroup.map f K
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) :
subgroup.map f (H ⊓ K) = subgroup.map f H ⊓ subgroup.map f K
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) :
add_subgroup.map f (H ⊓ K) = add_subgroup.map f H ⊓ add_subgroup.map f K
@[simp]
theorem
add_subgroup.map_bot
{G : Type u_1}
[add_group G]
{N : Type u_3}
[add_group N]
(f : G →+ N) :
add_subgroup.map f ⊥ = ⊥
@[simp]
theorem
subgroup.map_bot
{G : Type u_1}
[group G]
{N : Type u_3}
[group N]
(f : G →* N) :
subgroup.map f ⊥ = ⊥
@[simp]
theorem
subgroup.comap_top
{G : Type u_1}
[group G]
{N : Type u_3}
[group N]
(f : G →* N) :
subgroup.comap f ⊤ = ⊤
@[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
add_subgroup.comap_subtype_inf_left
{G : Type u_1}
[add_group G]
{H K : add_subgroup G} :
add_subgroup.comap H.subtype (H ⊓ K) = add_subgroup.comap H.subtype K
@[simp]
theorem
subgroup.comap_subtype_inf_left
{G : Type u_1}
[group G]
{H K : subgroup G} :
subgroup.comap H.subtype (H ⊓ K) = subgroup.comap H.subtype K
@[simp]
theorem
add_subgroup.comap_subtype_inf_right
{G : Type u_1}
[add_group G]
{H K : add_subgroup G} :
add_subgroup.comap K.subtype (H ⊓ K) = add_subgroup.comap K.subtype H
@[simp]
theorem
subgroup.comap_subtype_inf_right
{G : Type u_1}
[group G]
{H K : subgroup G} :
subgroup.comap K.subtype (H ⊓ K) = subgroup.comap K.subtype H
@[simp]
theorem
add_subgroup.comap_subtype_equiv_of_le_symm_apply_coe_coe
{G : Type u_1}
[add_group G]
{H K : add_subgroup G}
(h : H ≤ K)
(g : ↥H) :
@[simp]
theorem
add_subgroup.comap_subtype_equiv_of_le_apply_coe
{G : Type u_1}
[add_group G]
{H K : add_subgroup G}
(h : H ≤ K)
(g : ↥(add_subgroup.comap K.subtype H)) :
def
subgroup.comap_subtype_equiv_of_le
{G : Type u_1}
[group G]
{H K : subgroup G}
(h : H ≤ K) :
↥(subgroup.comap K.subtype H) ≃* ↥H
If H ≤ K, then H as a subgroup of K is isomorphic to H.
def
add_subgroup.comap_subtype_equiv_of_le
{G : Type u_1}
[add_group G]
{H K : add_subgroup G}
(h : H ≤ K) :
↥(add_subgroup.comap K.subtype H) ≃+ ↥H
If H ≤ K, then H as a subgroup of K is isomorphic to H.
@[simp]
theorem
subgroup.comap_subtype_equiv_of_le_apply_coe
{G : Type u_1}
[group G]
{H K : subgroup G}
(h : H ≤ K)
(g : ↥(subgroup.comap K.subtype H)) :
For any subgroups H and K, view H ⊓ K as a subgroup of K.
Equations
- H.add_subgroup_of K = add_subgroup.comap K.subtype H
For any subgroups H and K, view H ⊓ K as a subgroup of K.
Equations
- H.subgroup_of K = subgroup.comap K.subtype H
theorem
subgroup.mem_subgroup_of
{G : Type u_1}
[group G]
{H K : subgroup G}
{h : ↥K} :
h ∈ H.subgroup_of K ↔ ↑h ∈ H
theorem
add_subgroup.mem_add_subgroup_of
{G : Type u_1}
[add_group G]
{H K : add_subgroup G}
{h : ↥K} :
h ∈ H.add_subgroup_of K ↔ ↑h ∈ H
theorem
add_subgroup.add_subgroup_of_map_subtype
{G : Type u_1}
[add_group G]
(H K : add_subgroup G) :
add_subgroup.map K.subtype (H.add_subgroup_of K) = H ⊓ K
theorem
subgroup.subgroup_of_map_subtype
{G : Type u_1}
[group G]
(H K : subgroup G) :
subgroup.map K.subtype (H.subgroup_of K) = H ⊓ K
@[simp]
theorem
add_subgroup.bot_add_subgroup_of
{G : Type u_1}
[add_group G]
(H : add_subgroup G) :
⊥.add_subgroup_of H = ⊥
@[simp]
@[simp]
@[simp]
theorem
add_subgroup.top_add_subgroup_of
{G : Type u_1}
[add_group G]
(H : add_subgroup G) :
⊤.add_subgroup_of H = ⊤
theorem
add_subgroup.add_subgroup_of_bot_eq_bot
{G : Type u_1}
[add_group G]
(H : add_subgroup G) :
H.add_subgroup_of ⊥ = ⊥
theorem
subgroup.subgroup_of_bot_eq_bot
{G : Type u_1}
[group G]
(H : subgroup G) :
H.subgroup_of ⊥ = ⊥
theorem
add_subgroup.add_subgroup_of_bot_eq_top
{G : Type u_1}
[add_group G]
(H : add_subgroup G) :
H.add_subgroup_of ⊥ = ⊤
theorem
subgroup.subgroup_of_bot_eq_top
{G : Type u_1}
[group G]
(H : subgroup G) :
H.subgroup_of ⊥ = ⊤
@[simp]
@[simp]
theorem
add_subgroup.add_subgroup_of_self
{G : Type u_1}
[add_group G]
(H : add_subgroup G) :
H.add_subgroup_of H = ⊤
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) :
add_subgroup (G × N)
Given add_subgroups H, K of add_groups A, B respectively, H × K
as an add_subgroup of A × B.
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) :
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} :
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
add_subgroup.prod_top
{G : Type u_1}
[add_group G]
{N : Type u_3}
[add_group N]
(K : add_subgroup G) :
K.prod ⊤ = add_subgroup.comap (add_monoid_hom.fst G N) K
theorem
subgroup.prod_top
{G : Type u_1}
[group G]
{N : Type u_3}
[group N]
(K : subgroup G) :
K.prod ⊤ = subgroup.comap (monoid_hom.fst G N) K
theorem
add_subgroup.top_prod
{G : Type u_1}
[add_group G]
{N : Type u_3}
[add_group N]
(H : add_subgroup N) :
⊤.prod H = add_subgroup.comap (add_monoid_hom.snd G N) H
theorem
subgroup.top_prod
{G : Type u_1}
[group G]
{N : Type u_3}
[group N]
(H : subgroup N) :
⊤.prod H = subgroup.comap (monoid_hom.snd G N) H
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.
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
@[class]
A subgroup is normal if whenever n ∈ H, then g * n * g⁻¹ ∈ H for every g : G
Instances
- subgroup.normal_of_comm
- subgroup.normal_of_characteristic
- subgroup.normal_in_normalizer
- subgroup.normal_comap
- monoid_hom.normal_ker
- subgroup.normal_inf
- subgroup.normal_closure_normal
- subgroup.normal_core_normal
- subgroup.prod_subgroup_of_prod_normal
- subgroup.prod_normal
- subgroup.sup_normal
- subgroup.normal_inf_normal
- quotient_group.map_normal
@[class]
An add_subgroup is normal if whenever n ∈ H, then g + n - g ∈ H for every g : G
Instances
- add_subgroup.normal_of_comm
- add_subgroup.normal_of_characteristic
- add_subgroup.normal_in_normalizer
- add_subgroup.normal_comap
- add_monoid_hom.normal_ker
- add_subgroup.normal_inf
- add_subgroup.sum_add_subgroup_of_sum_normal
- add_subgroup.sum_normal
- add_subgroup.normal_inf_normal
- quotient_add_group.map_normal
@[protected, instance]
@[protected, instance]
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) :
theorem
add_subgroup.normal.mem_comm_iff
{G : Type u_1}
[add_group G]
{H : add_subgroup G}
(nH : H.normal)
{a b : G} :
@[class]
- fixed : ∀ (ϕ : G ≃* G), subgroup.comap ϕ.to_monoid_hom H = H
A subgroup is characteristic if it is fixed by all automorphisms.
Several equivalent conditions are provided by lemmas of the form characteristic.iff...
@[protected, instance]
def
subgroup.normal_of_characteristic
{G : Type u_1}
[group G]
(H : subgroup G)
[h : H.characteristic] :
H.normal
@[class]
- fixed : ∀ (ϕ : A ≃+ A), add_subgroup.comap ϕ.to_add_monoid_hom H = H
A add_subgroup is characteristic if it is fixed by all automorphisms.
Several equivalent conditions are provided by lemmas of the form characteristic.iff...
@[protected, instance]
def
add_subgroup.normal_of_characteristic
{A : Type u_2}
[add_group A]
(H : add_subgroup A)
[h : H.characteristic] :
H.normal
theorem
add_subgroup.characteristic_iff_comap_eq
{G : Type u_1}
[add_group G]
{H : add_subgroup G} :
H.characteristic ↔ ∀ (ϕ : G ≃+ G), add_subgroup.comap ϕ.to_add_monoid_hom H = H
theorem
subgroup.characteristic_iff_comap_eq
{G : Type u_1}
[group G]
{H : subgroup G} :
H.characteristic ↔ ∀ (ϕ : G ≃* G), subgroup.comap ϕ.to_monoid_hom H = H
theorem
add_subgroup.characteristic_iff_comap_le
{G : Type u_1}
[add_group G]
{H : add_subgroup G} :
H.characteristic ↔ ∀ (ϕ : G ≃+ G), add_subgroup.comap ϕ.to_add_monoid_hom H ≤ H
theorem
subgroup.characteristic_iff_comap_le
{G : Type u_1}
[group G]
{H : subgroup G} :
H.characteristic ↔ ∀ (ϕ : G ≃* G), subgroup.comap ϕ.to_monoid_hom H ≤ H
theorem
add_subgroup.characteristic_iff_le_comap
{G : Type u_1}
[add_group G]
{H : add_subgroup G} :
H.characteristic ↔ ∀ (ϕ : G ≃+ G), H ≤ add_subgroup.comap ϕ.to_add_monoid_hom H
theorem
subgroup.characteristic_iff_le_comap
{G : Type u_1}
[group G]
{H : subgroup G} :
H.characteristic ↔ ∀ (ϕ : G ≃* G), H ≤ subgroup.comap ϕ.to_monoid_hom H
theorem
subgroup.characteristic_iff_map_eq
{G : Type u_1}
[group G]
{H : subgroup G} :
H.characteristic ↔ ∀ (ϕ : G ≃* G), subgroup.map ϕ.to_monoid_hom H = H
theorem
add_subgroup.characteristic_iff_map_eq
{G : Type u_1}
[add_group G]
{H : add_subgroup G} :
H.characteristic ↔ ∀ (ϕ : G ≃+ G), add_subgroup.map ϕ.to_add_monoid_hom H = H
theorem
subgroup.characteristic_iff_map_le
{G : Type u_1}
[group G]
{H : subgroup G} :
H.characteristic ↔ ∀ (ϕ : G ≃* G), subgroup.map ϕ.to_monoid_hom H ≤ H
theorem
add_subgroup.characteristic_iff_map_le
{G : Type u_1}
[add_group G]
{H : add_subgroup G} :
H.characteristic ↔ ∀ (ϕ : G ≃+ G), add_subgroup.map ϕ.to_add_monoid_hom H ≤ H
theorem
add_subgroup.characteristic_iff_le_map
{G : Type u_1}
[add_group G]
{H : add_subgroup G} :
H.characteristic ↔ ∀ (ϕ : G ≃+ G), H ≤ add_subgroup.map ϕ.to_add_monoid_hom H
theorem
subgroup.characteristic_iff_le_map
{G : Type u_1}
[group G]
{H : subgroup G} :
H.characteristic ↔ ∀ (ϕ : G ≃* G), H ≤ subgroup.map ϕ.to_monoid_hom H
@[protected, instance]
@[protected, instance]
@[protected, instance]
@[protected, instance]
The center of an additive group G is the set of elements that commute with
everything in G
Equations
- add_subgroup.center G = {carrier := set.add_center G (add_zero_class.to_has_add G), zero_mem' := _, add_mem' := _, neg_mem' := _}
The center of a group G is the set of elements that commute with everything in G
Equations
- subgroup.center G = {carrier := set.center G (mul_one_class.to_has_mul G), one_mem' := _, mul_mem' := _, inv_mem' := _}
@[simp]
@[simp]
@[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
- subgroup.decidable_mem_center = λ (_x : G), decidable_of_iff' (∀ (g : G), g * _x = _x * g) subgroup.mem_center_iff
@[protected, instance]
@[protected, instance]
The normalizer of H is the largest subgroup of G inside which H is normal.
The set_normalizer of S is the subgroup of G whose elements satisfy g*S*g⁻¹=S
The set_normalizer of S is the subgroup of G whose elements satisfy
g+S-g=S.
theorem
add_subgroup.le_normalizer
{G : Type u_1}
[add_group G]
{H : add_subgroup G} :
H ≤ H.normalizer
@[protected, instance]
@[protected, instance]
theorem
add_subgroup.normalizer_eq_top
{G : Type u_1}
[add_group G]
{H : add_subgroup G} :
H.normalizer = ⊤ ↔ H.normal
theorem
subgroup.normalizer_eq_top
{G : Type u_1}
[group G]
{H : subgroup G} :
H.normalizer = ⊤ ↔ H.normal
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) :
K ≤ H.normalizer
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) :
K ≤ H.normalizer
theorem
subgroup.le_normalizer_comap
{G : Type u_1}
[group G]
{H : subgroup G}
{N : Type u_3}
[group N]
(f : N →* G) :
subgroup.comap f H.normalizer ≤ (subgroup.comap f H).normalizer
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) :
subgroup.map f H.normalizer ≤ (subgroup.map f H).normalizer
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) :
add_subgroup.map f H.normalizer ≤ (add_subgroup.map f H).normalizer
The image of the normalizer is contained in the normalizer of the image.
@[protected, instance]
def
add_subgroup.is_commutative.add_comm_group
{G : Type u_1}
[add_group G]
(H : add_subgroup G)
[h : H.is_commutative] :
Equations
- add_subgroup.is_commutative.add_comm_group H = {add := add_group.add H.to_add_group, add_assoc := _, zero := add_group.zero H.to_add_group, zero_add := _, add_zero := _, nsmul := add_group.nsmul H.to_add_group, nsmul_zero' := _, nsmul_succ' := _, neg := add_group.neg H.to_add_group, sub := add_group.sub H.to_add_group, sub_eq_add_neg := _, zsmul := add_group.zsmul H.to_add_group, zsmul_zero' := _, zsmul_succ' := _, zsmul_neg' := _, add_left_neg := _, add_comm := _}
@[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
- subgroup.is_commutative.comm_group H = {mul := group.mul H.to_group, mul_assoc := _, one := group.one H.to_group, one_mul := _, mul_one := _, npow := group.npow H.to_group, npow_zero' := _, npow_succ' := _, inv := group.inv H.to_group, div := group.div H.to_group, div_eq_mul_inv := _, zpow := group.zpow H.to_group, zpow_zero' := _, zpow_succ' := _, zpow_neg' := _, mul_left_inv := _, mul_comm := _}
@[protected, instance]
Given a set s, conjugates_of_set s is the set of all conjugates of
the elements of s.
Equations
- group.conjugates_of_set s = ⋃ (a : G) (H : a ∈ s), conjugates_of a
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.conjugates_subset_normal
{G : Type u_1}
[group G]
{N : subgroup G}
[tn : N.normal]
{a : G}
(h : a ∈ N) :
conjugates_of a ⊆ ↑N
theorem
group.conj_mem_conjugates_of_set
{G : Type u_1}
[group G]
{s : set G}
{x c : G} :
x ∈ group.conjugates_of_set s → (c * x) * c⁻¹ ∈ group.conjugates_of_set s
The set of conjugates of s is closed under conjugation.
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} :
s ⊆ ↑(subgroup.normal_closure s)
@[protected, instance]
The normal closure of s is a normal subgroup.
@[simp]
@[simp]
@[simp]
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.
@[protected, instance]
theorem
subgroup.normal_le_normal_core
{G : Type u_1}
[group G]
{H N : subgroup G}
[hN : N.normal] :
N ≤ H.normal_core ↔ N ≤ H
theorem
subgroup.normal_core_mono
{G : Type u_1}
[group G]
{H K : subgroup G}
(h : H ≤ K) :
H.normal_core ≤ K.normal_core
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] :
H.normal_core = H
@[simp]
The range of an add_monoid_hom from an add_group is an add_subgroup.
@[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
- f.decidable_mem_range = λ (x : N), fintype.decidable_exists_fintype
@[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
- f.decidable_mem_range = λ (x : N), fintype.decidable_exists_fintype
theorem
monoid_hom.range_eq_map
{G : Type u_1}
[group G]
{N : Type u_3}
[group N]
(f : G →* N) :
f.range = subgroup.map f ⊤
theorem
add_monoid_hom.range_eq_map
{G : Type u_1}
[add_group G]
{N : Type u_3}
[add_group N]
(f : G →+ N) :
f.range = add_subgroup.map f ⊤
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
- f.range_restrict = add_monoid_hom.mk' (λ (g : G), ⟨⇑f g, _⟩) _
@[simp]
theorem
monoid_hom.coe_range_restrict
{G : Type u_1}
[group G]
{N : Type u_3}
[group N]
(f : G →* N)
(g : G) :
↑(⇑(f.range_restrict) g) = ⇑f 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) :
↑(⇑(f.range_restrict) g) = ⇑f 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.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]
@[simp]
theorem
add_subgroup.inclusion_range
{G : Type u_1}
[add_group G]
{H K : add_subgroup G}
(h_le : H ≤ K) :
(add_subgroup.inclusion h_le).range = H.add_subgroup_of K
@[simp]
theorem
subgroup.inclusion_range
{G : Type u_1}
[group G]
{H K : subgroup G}
(h_le : H ≤ K) :
(subgroup.inclusion h_le).range = H.subgroup_of 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.
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.
@[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) :
f.range.subgroup_of K = (f.cod_restrict K _).range
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) :
f.range.add_subgroup_of K = (f.cod_restrict K _).range
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.
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.
@[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) :
↑(⇑(monoid_hom.of_left_inverse h) x) = ⇑f x
@[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) :
↑(⇑(add_monoid_hom.of_left_inverse h) x) = ⇑f x
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
- monoid_hom.of_injective hf = mul_equiv.of_bijective (f.cod_restrict f.range monoid_hom.of_injective._proof_1) _
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
- add_monoid_hom.of_injective hf = add_equiv.of_bijective (f.cod_restrict f.range add_monoid_hom.of_injective._proof_1) _
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} :
↑(⇑(monoid_hom.of_injective hf) x) = ⇑f x
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} :
↑(⇑(add_monoid_hom.of_injective hf) x) = ⇑f x
def
monoid_hom.ker
{G : Type u_1}
[group G]
{M : Type u_5}
[mul_one_class M]
(f : G →* M) :
subgroup G
The multiplicative kernel of a monoid homomorphism is the subgroup of elements x : G such that
f x = 1
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
theorem
monoid_hom.mem_ker
{G : Type u_1}
[group G]
{M : Type u_5}
[mul_one_class M]
(f : G →* M)
{x : G} :
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} :
theorem
add_monoid_hom.coe_ker
{G : Type u_1}
[add_group G]
{M : Type u_5}
[add_zero_class M]
(f : G →+ M) :
@[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
- f.decidable_mem_ker = λ (x : G), decidable_of_iff (⇑f x = 1) _
@[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
- f.decidable_mem_ker = λ (x : G), decidable_of_iff (⇑f x = 0) _
@[simp]
theorem
monoid_hom.comap_bot
{G : Type u_1}
[group G]
{N : Type u_3}
[group N]
(f : G →* N) :
subgroup.comap f ⊥ = f.ker
@[simp]
theorem
add_monoid_hom.comap_bot
{G : Type u_1}
[add_group G]
{N : Type u_3}
[add_group N]
(f : G →+ N) :
add_subgroup.comap f ⊥ = f.ker
theorem
add_monoid_hom.range_restrict_ker
{G : Type u_1}
[add_group G]
{N : Type u_3}
[add_group N]
(f : G →+ N) :
f.range_restrict.ker = f.ker
theorem
monoid_hom.range_restrict_ker
{G : Type u_1}
[group G]
{N : Type u_3}
[group N]
(f : G →* N) :
f.range_restrict.ker = f.ker
@[simp]
@[simp]
@[simp]
@[simp]
theorem
subgroup.ker_inclusion
{G : Type u_1}
[group G]
{H K : subgroup G}
(h : H ≤ K) :
(subgroup.inclusion h).ker = ⊥
@[simp]
theorem
add_subgroup.ker_inclusion
{G : Type u_1}
[add_group G]
{H K : add_subgroup G}
(h : H ≤ K) :
(add_subgroup.inclusion h).ker = ⊥
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') :
add_subgroup.comap (f.prod_map g) (S.prod S') = (add_subgroup.comap f S).prod (add_subgroup.comap g S')
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') :
subgroup.comap (f.prod_map g) (S.prod S') = (subgroup.comap f S).prod (subgroup.comap g S')
theorem
monoid_hom.gclosure_preimage_le
{G : Type u_1}
[group G]
{N : Type u_3}
[group N]
(f : G →* N)
(s : set N) :
subgroup.closure (⇑f ⁻¹' s) ≤ subgroup.comap f (subgroup.closure s)
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) :
add_subgroup.closure (⇑f ⁻¹' s) ≤ add_subgroup.comap f (add_subgroup.closure s)
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) :
add_subgroup.map f (add_subgroup.closure s) = add_subgroup.closure (⇑f '' s)
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) :
subgroup.map f (subgroup.closure s) = subgroup.closure (⇑f '' s)
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
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) :
add_subgroup.map f H ≤ f.range
theorem
subgroup.map_le_range
{G : Type u_1}
[group G]
{N : Type u_3}
[group N]
(f : G →* N)
(H : subgroup G) :
subgroup.map f H ≤ f.range
theorem
subgroup.map_subtype_le
{G : Type u_1}
[group G]
{H : subgroup G}
(K : subgroup ↥H) :
subgroup.map H.subtype K ≤ H
theorem
add_subgroup.map_subtype_le
{G : Type u_1}
[add_group G]
{H : add_subgroup G}
(K : add_subgroup ↥H) :
add_subgroup.map H.subtype K ≤ H
theorem
subgroup.ker_le_comap
{G : Type u_1}
[group G]
{N : Type u_3}
[group N]
(f : G →* N)
(H : subgroup N) :
f.ker ≤ subgroup.comap f H
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) :
f.ker ≤ add_subgroup.comap f H
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) :
add_subgroup.map f (add_subgroup.comap f H) ≤ H
theorem
subgroup.map_comap_le
{G : Type u_1}
[group G]
{N : Type u_3}
[group N]
(f : G →* N)
(H : subgroup N) :
subgroup.map f (subgroup.comap f H) ≤ H
theorem
subgroup.le_comap_map
{G : Type u_1}
[group G]
{N : Type u_3}
[group N]
(f : G →* N)
(H : subgroup G) :
H ≤ subgroup.comap f (subgroup.map f H)
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) :
H ≤ add_subgroup.comap f (add_subgroup.map f H)
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) :
add_subgroup.map f (add_subgroup.comap f H) = f.range ⊓ H
theorem
subgroup.map_comap_eq
{G : Type u_1}
[group G]
{N : Type u_3}
[group N]
(f : G →* N)
(H : subgroup N) :
subgroup.map f (subgroup.comap f H) = f.range ⊓ H
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) :
add_subgroup.comap f (add_subgroup.map f H) = H ⊔ f.ker
theorem
subgroup.comap_map_eq
{G : Type u_1}
[group G]
{N : Type u_3}
[group N]
(f : G →* N)
(H : subgroup G) :
subgroup.comap f (subgroup.map f H) = H ⊔ f.ker
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) :
add_subgroup.map f (add_subgroup.comap f H) = H
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) :
subgroup.map f (subgroup.comap f H) = H
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) :
add_subgroup.map f (add_subgroup.comap f H) = H
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) :
subgroup.map f (subgroup.comap f H) = H
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) :
subgroup.comap f (subgroup.map f H) = 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) :
add_subgroup.comap f (add_subgroup.map f H) = 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) :
subgroup.comap f (subgroup.map f H) = H
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) :
add_subgroup.comap f (add_subgroup.map f H) = H
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) :
subgroup.map f H = subgroup.comap g H
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) :
add_subgroup.map f H = add_subgroup.comap g H
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.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) :
subgroup.comap f H ⊔ subgroup.comap f K = subgroup.comap f (H ⊔ K)
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) :
add_subgroup.comap f H ⊔ add_subgroup.comap f K = add_subgroup.comap f (H ⊔ K)
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) :
add_subgroup.comap f H ⊔ add_subgroup.comap f K = add_subgroup.comap f (H ⊔ K)
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) :
subgroup.comap f H ⊔ subgroup.comap f K = subgroup.comap f (H ⊔ K)
theorem
subgroup.sup_subgroup_of_eq
{G : Type u_1}
[group G]
{H K L : subgroup G}
(hH : H ≤ L)
(hK : K ≤ L) :
H.subgroup_of L ⊔ K.subgroup_of L = (H ⊔ K).subgroup_of 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) :
H.add_subgroup_of L ⊔ K.add_subgroup_of L = (H ⊔ K).add_subgroup_of 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) :
↥H ≃* ↥(subgroup.map f H)
A subgroup is isomorphic to its image under an injective function
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) :
↥H ≃+ ↥(add_subgroup.map f H)
An additive subgroup is isomorphic to its image under an injective function
@[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) :
theorem
add_subgroup.comap_normalizer_eq_of_surjective
{G : Type u_1}
[add_group G]
{N : Type u_3}
[add_group N]
(H : add_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.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) :
subgroup.comap f H.normalizer = (subgroup.comap f H).normalizer
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.
theorem
add_subgroup.map_equiv_normalizer_eq
{G : Type u_1}
[add_group G]
{N : Type u_3}
[add_group N]
(H : add_subgroup G)
(f : G ≃+ N) :
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) :
subgroup.map f H.normalizer = (subgroup.map f H).normalizer
The image of the normalizer is equal to the normalizer of the image of a bijective function.
theorem
add_subgroup.map_normalizer_eq_of_bijective
{G : Type u_1}
[add_group G]
{N : Type u_3}
[add_group N]
(H : add_subgroup G)
{f : G →+ N}
(hf : function.bijective ⇑f) :
add_subgroup.map f H.normalizer = (add_subgroup.map f H).normalizer
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
@[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) :
lift_of_right_inverse f f_inv hf g hg is the unique additive group homomorphism φ
- such that
φ.comp f = g(add_monoid_hom.lift_of_right_inverse_comp), - where
f : G₁ →+ G₂has a right_inversef_inv(hf), - and
g : G₂ →+ G₃satisfieshg : f.ker ≤ g.ker.
See add_monoid_hom.eq_lift_of_right_inverse for the uniqueness lemma.
G₁.
| \
f | \ g
| \
v \⌟
G₂----> G₃
∃!φ
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) :
lift_of_right_inverse f hf g hg is the unique group homomorphism φ
- such that
φ.comp f = g(monoid_hom.lift_of_right_inverse_comp), - where
f : G₁ →+* G₂has a right_inversef_inv(hf), - and
g : G₂ →+* G₃satisfieshg : f.ker ≤ g.ker.
See monoid_hom.eq_lift_of_right_inverse for the uniqueness lemma.
G₁.
| \
f | \ g
| \
v \⌟
G₂----> G₃
∃!φ
@[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) :
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) :
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]
@[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) :
@[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
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) :
(subgroup.comap f H).normal
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) :
(add_subgroup.comap f H).normal
@[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) :
(subgroup.comap f H).normal
@[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) :
(add_subgroup.comap f H).normal
@[protected, instance]
def
subgroup.normal_inf
{G : Type u_1}
[group G]
(H N : subgroup G)
[hN : N.normal] :
(subgroup.comap H.subtype (H ⊓ N)).normal
@[protected, instance]
def
add_subgroup.normal_inf
{G : Type u_1}
[add_group G]
(H N : add_subgroup G)
[hN : N.normal] :
(add_subgroup.comap H.subtype (H ⊓ N)).normal
The subgroup generated by an element.
Equations
- subgroup.zpowers g = (⇑(zpowers_hom G) g).range.copy (set.range (pow g)) _
@[simp]
@[simp]
theorem
subgroup.range_zpowers_hom
{G : Type u_1}
[group G]
(g : G) :
(⇑(zpowers_hom G) g).range = subgroup.zpowers g
theorem
subgroup.zpowers_subset
{G : Type u_1}
[group G]
{a : G}
{K : subgroup G}
(h : a ∈ K) :
subgroup.zpowers a ≤ K
@[simp]
theorem
subgroup.forall_zpowers
{G : Type u_1}
[group G]
{x : G}
{p : ↥(subgroup.zpowers x) → Prop} :
@[simp]
theorem
subgroup.exists_zpowers
{G : Type u_1}
[group G]
{x : G}
{p : ↥(subgroup.zpowers x) → Prop} :
theorem
subgroup.forall_mem_zpowers
{G : Type u_1}
[group G]
{x : G}
{p : G → Prop} :
(∀ (g : G), g ∈ subgroup.zpowers x → p g) ↔ ∀ (m : ℤ), 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 : ℤ), p (x ^ m)
The subgroup generated by an element.
Equations
- add_subgroup.zmultiples a = (⇑(zmultiples_hom A) a).range.copy (set.range (λ (_x : ℤ), _x • a)) _
@[simp]
theorem
add_subgroup.range_zmultiples_hom
{A : Type u_2}
[add_group A]
(a : A) :
(⇑(zmultiples_hom A) a).range = add_subgroup.zmultiples a
@[simp]
theorem
add_subgroup.zmultiples_subset
{G : Type u_1}
[add_group G]
{a : G}
{K : add_subgroup G}
(h : a ∈ K) :
@[simp]
theorem
add_subgroup.forall_zmultiples
{G : Type u_1}
[add_group G]
{x : G}
{p : ↥(add_subgroup.zmultiples x) → Prop} :
theorem
add_subgroup.forall_mem_zmultiples
{G : Type u_1}
[add_group G]
{x : G}
{p : G → Prop} :
(∀ (g : G), g ∈ add_subgroup.zmultiples x → p g) ↔ ∀ (m : ℤ), p (m • x)
@[simp]
theorem
add_subgroup.exists_zmultiples
{G : Type u_1}
[add_group G]
{x : G}
{p : ↥(add_subgroup.zmultiples x) → Prop} :
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 : ℤ), 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') :
↥(subgroup.comap f H') →* ↥H'
The monoid_hom from the preimage of a subgroup to itself.
Equations
- f.subgroup_comap H' = f.submonoid_comap H'.to_submonoid
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') :
↥(add_subgroup.comap f H') →+ ↥H'
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) :
↥H →* ↥(subgroup.map f H)
The monoid_hom from a subgroup to its image.
Equations
- f.subgroup_map H = f.submonoid_map H.to_submonoid
@[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) :
↥H →+ ↥(add_subgroup.map f H)
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) :
Makes the identity isomorphism from a proof two subgroups of a multiplicative group are equal.
Equations
- mul_equiv.subgroup_congr h = {to_fun := (equiv.set_congr _).to_fun, inv_fun := (equiv.set_congr _).inv_fun, left_inv := _, right_inv := _, map_mul' := _}
Makes the identity additive isomorphism from a proof two subgroups of an additive group are equal.
Equations
- add_equiv.add_subgroup_congr h = {to_fun := (equiv.set_congr _).to_fun, inv_fun := (equiv.set_congr _).inv_fun, left_inv := _, right_inv := _, map_add' := _}
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) :
↥H ≃+ ↥(add_subgroup.map e.to_add_monoid_hom H)
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) :
↥H ≃* ↥(subgroup.map e.to_monoid_hom H)
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
@[protected, instance]
@[protected, instance]
@[instance]
@[class]
- exists_pair_ne : ∃ (x y : A), x ≠ y
- eq_bot_or_eq_top_of_normal : ∀ (H : add_subgroup A), H.normal → H = ⊥ ∨ H = ⊤
An add_group is simple when it has exactly two normal add_subgroups.
@[instance]
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) :
@[protected, instance]
def
is_simple_add_group.subgroup.is_simple_order
{C : Type u_1}
[add_comm_group C]
[is_simple_add_group C] :
@[protected, instance]
theorem
is_simple_add_group.is_simple_add_group_of_surjective
{G : Type u_1}
[add_group G]
{H : Type u_2}
[add_group H]
[is_simple_add_group G]
[nontrivial H]
(f : G →+ H)
(hf : function.surjective ⇑f) :
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.closure_mul_le
{G : Type u_1}
[group G]
(S T : set G) :
subgroup.closure (S * T) ≤ subgroup.closure S ⊔ subgroup.closure T
theorem
add_subgroup.normal_add_subgroup_of_iff
{G : Type u_1}
[add_group G]
{H K : add_subgroup G}
(hHK : H ≤ K) :
@[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] :
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
add_subgroup.normal_inf_normal
{G : Type u_1}
[add_group G]
(H K : add_subgroup G)
[hH : H.normal]
[hK : 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) :
(A ⊔ A').add_subgroup_of B = A.add_subgroup_of B ⊔ A'.add_subgroup_of B
theorem
subgroup.subgroup_of_sup
{G : Type u_1}
[group G]
(A A' B : subgroup G)
(hA : A ≤ B)
(hA' : A' ≤ B) :
(A ⊔ A').subgroup_of B = A.subgroup_of B ⊔ A'.subgroup_of 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) :
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) :
add_action ↥S α
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) :
mul_action ↥S α
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) :
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) :
@[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) :
smul_comm_class ↥S α β
@[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) :
vadd_comm_class ↥S α β
@[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) :
smul_comm_class α ↥S β
@[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) :
vadd_comm_class α ↥S β
@[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) :
is_scalar_tower ↥S α β
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 #
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 ℕ A₂]
(f : A₁ →+ A₂) :