mathlib documentation

group_theory.coset

Cosets #

This file develops the basic theory of left and right cosets.

Main definitions #

Notation #

TODO #

Add to_additive to preimage_mk_equiv_subgroup_times_set.

def left_add_coset {α : Type u_1} [has_add α] (a : α) (s : set α) :
set α

The left coset a+s for an element a : α and a subset s : set α

def left_coset {α : Type u_1} [has_mul α] (a : α) (s : set α) :
set α

The left coset a * s for an element a : α and a subset s : set α

Equations
  • a *l s = (λ (x : α), a * x) '' s
def right_coset {α : Type u_1} [has_mul α] (s : set α) (a : α) :
set α

The right coset s * a for an element a : α and a subset s : set α

Equations
  • s *r a = (λ (x : α), x * a) '' s
def right_add_coset {α : Type u_1} [has_add α] (s : set α) (a : α) :
set α

The right coset s+a for an element a : α and a subset s : set α

theorem mem_left_coset {α : Type u_1} [has_mul α] {s : set α} {x : α} (a : α) (hxS : x s) :
a * x a *l s
theorem mem_left_add_coset {α : Type u_1} [has_add α] {s : set α} {x : α} (a : α) (hxS : x s) :
a + x a +l s
theorem mem_right_coset {α : Type u_1} [has_mul α] {s : set α} {x : α} (a : α) (hxS : x s) :
x * a s *r a
theorem mem_right_add_coset {α : Type u_1} [has_add α] {s : set α} {x : α} (a : α) (hxS : x s) :
x + a s +r a
def left_coset_equivalence {α : Type u_1} [has_mul α] (s : set α) (a b : α) :
Prop

Equality of two left cosets a * s and b * s.

Equations
def left_add_coset_equivalence {α : Type u_1} [has_add α] (s : set α) (a b : α) :
Prop

Equality of two left cosets a + s and b + s.

theorem left_coset_equivalence_rel {α : Type u_1} [has_mul α] (s : set α) :
def right_coset_equivalence {α : Type u_1} [has_mul α] (s : set α) (a b : α) :
Prop

Equality of two right cosets s * a and s * b.

Equations
def right_add_coset_equivalence {α : Type u_1} [has_add α] (s : set α) (a b : α) :
Prop

Equality of two right cosets s + a and s + b.

theorem right_coset_equivalence_rel {α : Type u_1} [has_mul α] (s : set α) :
@[simp]
theorem left_coset_assoc {α : Type u_1} [semigroup α] (s : set α) (a b : α) :
a *l (b *l s) = a * b *l s
@[simp]
theorem left_add_coset_assoc {α : Type u_1} [add_semigroup α] (s : set α) (a b : α) :
a +l (b +l s) = (a + b) +l s
@[simp]
theorem right_coset_assoc {α : Type u_1} [semigroup α] (s : set α) (a b : α) :
s *r a *r b = s *r a * b
@[simp]
theorem right_add_coset_assoc {α : Type u_1} [add_semigroup α] (s : set α) (a b : α) :
s +r a +r b = s +r (a + b)
theorem left_coset_right_coset {α : Type u_1} [semigroup α] (s : set α) (a b : α) :
a *l s *r b = a *l (s *r b)
theorem left_add_coset_right_add_coset {α : Type u_1} [add_semigroup α] (s : set α) (a b : α) :
a +l s +r b = a +l (s +r b)
@[simp]
theorem one_left_coset {α : Type u_1} [monoid α] (s : set α) :
1 *l s = s
@[simp]
theorem zero_left_add_coset {α : Type u_1} [add_monoid α] (s : set α) :
0 +l s = s
@[simp]
theorem right_coset_one {α : Type u_1} [monoid α] (s : set α) :
s *r 1 = s
@[simp]
theorem right_add_coset_zero {α : Type u_1} [add_monoid α] (s : set α) :
s +r 0 = s
theorem mem_own_left_coset {α : Type u_1} [monoid α] (s : submonoid α) (a : α) :
a a *l s
theorem mem_own_left_add_coset {α : Type u_1} [add_monoid α] (s : add_submonoid α) (a : α) :
a a +l s
theorem mem_own_right_coset {α : Type u_1} [monoid α] (s : submonoid α) (a : α) :
a s *r a
theorem mem_own_right_add_coset {α : Type u_1} [add_monoid α] (s : add_submonoid α) (a : α) :
a s +r a
theorem mem_left_add_coset_left_add_coset {α : Type u_1} [add_monoid α] (s : add_submonoid α) {a : α} (ha : a +l s = s) :
a s
theorem mem_left_coset_left_coset {α : Type u_1} [monoid α] (s : submonoid α) {a : α} (ha : a *l s = s) :
a s
theorem mem_right_coset_right_coset {α : Type u_1} [monoid α] (s : submonoid α) {a : α} (ha : s *r a = s) :
a s
theorem mem_right_add_coset_right_add_coset {α : Type u_1} [add_monoid α] (s : add_submonoid α) {a : α} (ha : s +r a = s) :
a s
theorem mem_left_add_coset_iff {α : Type u_1} [add_group α] {s : set α} {x : α} (a : α) :
x a +l s -a + x s
theorem mem_left_coset_iff {α : Type u_1} [group α] {s : set α} {x : α} (a : α) :
x a *l s a⁻¹ * x s
theorem mem_right_coset_iff {α : Type u_1} [group α] {s : set α} {x : α} (a : α) :
x s *r a x * a⁻¹ s
theorem mem_right_add_coset_iff {α : Type u_1} [add_group α] {s : set α} {x : α} (a : α) :
x s +r a x + -a s
theorem left_coset_mem_left_coset {α : Type u_1} [group α] (s : subgroup α) {a : α} (ha : a s) :
a *l s = s
theorem left_add_coset_mem_left_add_coset {α : Type u_1} [add_group α] (s : add_subgroup α) {a : α} (ha : a s) :
a +l s = s
theorem right_coset_mem_right_coset {α : Type u_1} [group α] (s : subgroup α) {a : α} (ha : a s) :
s *r a = s
theorem right_add_coset_mem_right_add_coset {α : Type u_1} [add_group α] (s : add_subgroup α) {a : α} (ha : a s) :
s +r a = s
theorem eq_cosets_of_normal {α : Type u_1} [group α] (s : subgroup α) (N : s.normal) (g : α) :
g *l s = s *r g
theorem eq_add_cosets_of_normal {α : Type u_1} [add_group α] (s : add_subgroup α) (N : s.normal) (g : α) :
g +l s = s +r g
theorem normal_of_eq_cosets {α : Type u_1} [group α] (s : subgroup α) (h : ∀ (g : α), g *l s = s *r g) :
theorem normal_of_eq_add_cosets {α : Type u_1} [add_group α] (s : add_subgroup α) (h : ∀ (g : α), g +l s = s +r g) :
theorem normal_iff_eq_add_cosets {α : Type u_1} [add_group α] (s : add_subgroup α) :
s.normal ∀ (g : α), g +l s = s +r g
theorem normal_iff_eq_cosets {α : Type u_1} [group α] (s : subgroup α) :
s.normal ∀ (g : α), g *l s = s *r g
theorem left_add_coset_eq_iff {α : Type u_1} [add_group α] (s : add_subgroup α) {x y : α} :
x +l s = y +l s -x + y s
theorem left_coset_eq_iff {α : Type u_1} [group α] (s : subgroup α) {x y : α} :
x *l s = y *l s x⁻¹ * y s
theorem right_coset_eq_iff {α : Type u_1} [group α] (s : subgroup α) {x y : α} :
s *r x = s *r y y * x⁻¹ s
theorem right_add_coset_eq_iff {α : Type u_1} [add_group α] (s : add_subgroup α) {x y : α} :
s +r x = s +r y y + -x s
def quotient_add_group.left_rel {α : Type u_1} [add_group α] (s : add_subgroup α) :

The equivalence relation corresponding to the partition of a group by left cosets of a subgroup.

def quotient_group.left_rel {α : Type u_1} [group α] (s : subgroup α) :

The equivalence relation corresponding to the partition of a group by left cosets of a subgroup.

Equations
@[instance]
def quotient_group.left_rel_decidable {α : Type u_1} [group α] (s : subgroup α) [decidable_pred (λ (_x : α), _x s)] :
Equations
@[instance]
def quotient_add_group.left_rel_decidable {α : Type u_1} [add_group α] (s : add_subgroup α) [decidable_pred (λ (_x : α), _x s)] :
def quotient_group.quotient {α : Type u_1} [group α] (s : subgroup α) :
Type u_1

quotient s is the quotient type representing the left cosets of s. If s is a normal subgroup, quotient s is a group

Equations
def quotient_group.right_rel {α : Type u_1} [group α] (s : subgroup α) :

The equivalence relation corresponding to the partition of a group by right cosets of a subgroup.

Equations
def quotient_add_group.right_rel {α : Type u_1} [add_group α] (s : add_subgroup α) :

The equivalence relation corresponding to the partition of a group by right cosets of a subgroup.

@[instance]
def quotient_group.right_rel_decidable {α : Type u_1} [group α] (s : subgroup α) [decidable_pred (λ (_x : α), _x s)] :
Equations
@[instance]
def quotient_add_group.right_rel_decidable {α : Type u_1} [add_group α] (s : add_subgroup α) [decidable_pred (λ (_x : α), _x s)] :
def quotient_add_group.quotient {α : Type u_1} [add_group α] (s : add_subgroup α) :
Type u_1

quotient s is the quotient type representing the left cosets of s. If s is a normal subgroup, quotient s is a group

Equations
def quotient_add_group.mk {α : Type u_1} [add_group α] {s : add_subgroup α} (a : α) :

The canonical map from an add_group α to the quotient α/s.

def quotient_group.mk {α : Type u_1} [group α] {s : subgroup α} (a : α) :

The canonical map from a group α to the quotient α/s.

theorem quotient_group.induction_on {α : Type u_1} [group α] {s : subgroup α} {C : quotient_group.quotient s → Prop} (x : quotient_group.quotient s) (H : ∀ (z : α), C (quotient_group.mk z)) :
C x
theorem quotient_add_group.induction_on {α : Type u_1} [add_group α] {s : add_subgroup α} {C : quotient_add_group.quotient s → Prop} (x : quotient_add_group.quotient s) (H : ∀ (z : α), C (quotient_add_group.mk z)) :
C x
theorem quotient_add_group.induction_on' {α : Type u_1} [add_group α] {s : add_subgroup α} {C : quotient_add_group.quotient s → Prop} (x : quotient_add_group.quotient s) (H : ∀ (z : α), C z) :
C x
theorem quotient_group.induction_on' {α : Type u_1} [group α] {s : subgroup α} {C : quotient_group.quotient s → Prop} (x : quotient_group.quotient s) (H : ∀ (z : α), C z) :
C x
theorem quotient_add_group.forall_coe {α : Type u_1} [add_group α] {s : add_subgroup α} {C : quotient_add_group.quotient s → Prop} :
(∀ (x : quotient_add_group.quotient s), C x) ∀ (x : α), C x
theorem quotient_group.forall_coe {α : Type u_1} [group α] {s : subgroup α} {C : quotient_group.quotient s → Prop} :
(∀ (x : quotient_group.quotient s), C x) ∀ (x : α), C x
theorem quotient_add_group.eq {α : Type u_1} [add_group α] {s : add_subgroup α} {a b : α} :
a = b -a + b s
theorem quotient_group.eq {α : Type u_1} [group α] {s : subgroup α} {a b : α} :
a = b a⁻¹ * b s
theorem quotient_add_group.eq' {α : Type u_1} [add_group α] {s : add_subgroup α} {a b : α} :
theorem quotient_group.eq' {α : Type u_1} [group α] {s : subgroup α} {a b : α} :
theorem quotient_group.out_eq' {α : Type u_1} [group α] {s : subgroup α} (a : quotient_group.quotient s) :
theorem quotient_add_group.mk_out'_eq_mul {α : Type u_1} [add_group α] (s : add_subgroup α) (g : α) :
theorem quotient_group.mk_out'_eq_mul {α : Type u_1} [group α] (s : subgroup α) (g : α) :
theorem quotient_add_group.mk_mul_of_mem {α : Type u_1} [add_group α] {s : add_subgroup α} (g₁ g₂ : α) (hg₂ : g₂ s) :
theorem quotient_group.mk_mul_of_mem {α : Type u_1} [group α] {s : subgroup α} (g₁ g₂ : α) (hg₂ : g₂ s) :
theorem quotient_group.eq_class_eq_left_coset {α : Type u_1} [group α] (s : subgroup α) (g : α) :
{x : α | x = g} = g *l s
theorem quotient_add_group.eq_class_eq_left_coset {α : Type u_1} [add_group α] (s : add_subgroup α) (g : α) :
{x : α | x = g} = g +l s
theorem quotient_group.preimage_image_coe {α : Type u_1} [group α] (N : subgroup α) (s : set α) :
coe ⁻¹' (coe '' s) = ⋃ (x : N), (λ (y : α), y * x) ⁻¹' s
theorem quotient_add_group.preimage_image_coe {α : Type u_1} [add_group α] (N : add_subgroup α) (s : set α) :
coe ⁻¹' (coe '' s) = ⋃ (x : N), (λ (y : α), y + x) ⁻¹' s
def subgroup.left_coset_equiv_subgroup {α : Type u_1} [group α] {s : subgroup α} (g : α) :

The natural bijection between a left coset g * s and s.

Equations
def add_subgroup.left_coset_equiv_add_subgroup {α : Type u_1} [add_group α] {s : add_subgroup α} (g : α) :

The natural bijection between the cosets g + s and s.

def subgroup.right_coset_equiv_subgroup {α : Type u_1} [group α] {s : subgroup α} (g : α) :

The natural bijection between a right coset s * g and s.

Equations
def add_subgroup.right_coset_equiv_add_subgroup {α : Type u_1} [add_group α] {s : add_subgroup α} (g : α) :

The natural bijection between the cosets s + g and s.

A (non-canonical) bijection between an add_group α and the product (α/s) × s

If H ≤ K, then G/H ≃ G/K × K/H constructively, using the provided right inverse of the quotient map G → G/K. The classical version is quotient_equiv_prod_of_le.

Equations

If H ≤ K, then G/H ≃ G/K × K/H constructively, using the provided right inverse of the quotient map G → G/K. The classical version is quotient_equiv_prod_of_le.

@[simp]
theorem subgroup.quotient_equiv_prod_of_le'_apply {α : Type u_1} [group α] {s t : subgroup α} (h_le : s t) (f : quotient_group.quotient t → α) (hf : function.right_inverse f quotient_group.mk) (a : quotient_group.quotient s) :
(subgroup.quotient_equiv_prod_of_le' h_le f hf) a = (quotient.map' id _ a, quotient.map' (λ (g : α), (f (quotient.mk' g))⁻¹ * g, _⟩) _ a)

If H ≤ K, then G/H ≃ G/K × K/H nonconstructively. The constructive version is quotient_equiv_prod_of_le'.

@[simp]
theorem add_subgroup.quotient_equiv_sum_of_le_apply {α : Type u_1} [add_group α] {s t : add_subgroup α} (h_le : s t) (a : quotient_add_group.quotient s) :

If H ≤ K, then G/H ≃ G/K × K/H nonconstructively. The constructive version is quotient_equiv_prod_of_le'.

Equations
@[simp]
theorem subgroup.quotient_equiv_prod_of_le_apply {α : Type u_1} [group α] {s t : subgroup α} (h_le : s t) (a : quotient_group.quotient s) :
theorem subgroup.card_subgroup_dvd_card {α : Type u_1} [group α] [fintype α] (s : subgroup α) [fintype s] :

Order of a Subgroup

theorem subgroup.card_quotient_dvd_card {α : Type u_1} [group α] [fintype α] (s : subgroup α) [decidable_pred (λ (a : α), a s)] [fintype s] :
theorem subgroup.card_dvd_of_injective {α : Type u_1} [group α] {H : Type u_2} [group H] [fintype α] [fintype H] (f : α →* H) (hf : function.injective f) :
theorem subgroup.card_dvd_of_le {α : Type u_1} [group α] {H K : subgroup α} [fintype H] [fintype K] (hHK : H K) :
theorem subgroup.card_comap_dvd_of_injective {α : Type u_1} [group α] {H : Type u_2} [group H] (K : subgroup H) [fintype K] (f : α →* H) [fintype (subgroup.comap f K)] (hf : function.injective f) :

If s is a subgroup of the group α, and t is a subset of α/s, then there is a (typically non-canonical) bijection between the preimage of t in α and the product s × t.

Equations