mathlib3 documentation

group_theory.coset

Cosets #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

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

Main definitions #

Notation #

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

Equations
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
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
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

Equations
theorem mem_left_coset {α : Type u_1} [has_mul α] {s : set α} {x : α} (a : α) (hxS : x s) :
a * x left_coset a s
theorem mem_left_add_coset {α : Type u_1} [has_add α] {s : set α} {x : α} (a : α) (hxS : x s) :
theorem mem_right_coset {α : Type u_1} [has_mul α] {s : set α} {x : α} (a : α) (hxS : x s) :
x * a right_coset s a
theorem mem_right_add_coset {α : Type u_1} [has_add α] {s : set α} {x : α} (a : α) (hxS : x s) :
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.

Equations
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.

Equations
@[simp]
theorem left_add_coset_assoc {α : Type u_1} [add_semigroup α] (s : set α) (a b : α) :
@[simp]
theorem left_coset_assoc {α : Type u_1} [semigroup α] (s : set α) (a b : α) :
@[simp]
theorem right_coset_assoc {α : Type u_1} [semigroup α] (s : set α) (a b : α) :
@[simp]
theorem right_add_coset_assoc {α : Type u_1} [add_semigroup α] (s : set α) (a b : α) :
theorem left_coset_right_coset {α : Type u_1} [semigroup α] (s : set α) (a b : α) :
@[simp]
theorem zero_left_add_coset {α : Type u_1} [add_monoid α] (s : set α) :
@[simp]
theorem one_left_coset {α : Type u_1} [monoid α] (s : set α) :
@[simp]
theorem right_add_coset_zero {α : Type u_1} [add_monoid α] (s : set α) :
@[simp]
theorem right_coset_one {α : Type u_1} [monoid α] (s : set α) :
theorem mem_own_left_coset {α : Type u_1} [monoid α] (s : submonoid α) (a : α) :
theorem mem_own_left_add_coset {α : Type u_1} [add_monoid α] (s : add_submonoid α) (a : α) :
theorem mem_own_right_coset {α : Type u_1} [monoid α] (s : submonoid α) (a : α) :
theorem mem_own_right_add_coset {α : Type u_1} [add_monoid α] (s : add_submonoid α) (a : α) :
theorem mem_left_add_coset_left_add_coset {α : Type u_1} [add_monoid α] (s : add_submonoid α) {a : α} (ha : left_add_coset a s = s) :
a s
theorem mem_left_coset_left_coset {α : Type u_1} [monoid α] (s : submonoid α) {a : α} (ha : left_coset a s = s) :
a s
theorem mem_right_coset_right_coset {α : Type u_1} [monoid α] (s : submonoid α) {a : α} (ha : right_coset s 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 : right_add_coset s a = s) :
a s
theorem mem_left_add_coset_iff {α : Type u_1} [add_group α] {s : set α} {x : α} (a : α) :
theorem mem_left_coset_iff {α : Type u_1} [group α] {s : set α} {x : α} (a : α) :
theorem mem_right_coset_iff {α : Type u_1} [group α] {s : set α} {x : α} (a : α) :
theorem mem_right_add_coset_iff {α : Type u_1} [add_group α] {s : set α} {x : α} (a : α) :
theorem left_coset_mem_left_coset {α : Type u_1} [group α] (s : subgroup α) {a : α} (ha : a 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) :
theorem right_coset_mem_right_coset {α : Type u_1} [group α] (s : subgroup α) {a : α} (ha : 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) :
theorem orbit_subgroup_eq_right_coset {α : Type u_1} [group α] (s : subgroup α) (a : α) :
theorem orbit_add_subgroup_eq_self_of_mem {α : Type u_1} [add_group α] (s : add_subgroup α) {a : α} (ha : a s) :
theorem orbit_subgroup_eq_self_of_mem {α : Type u_1} [group α] (s : subgroup α) {a : α} (ha : a s) :
theorem orbit_subgroup_one_eq_self {α : Type u_1} [group α] (s : subgroup α) :
theorem eq_cosets_of_normal {α : Type u_1} [group α] (s : subgroup α) (N : s.normal) (g : α) :
theorem eq_add_cosets_of_normal {α : Type u_1} [add_group α] (s : add_subgroup α) (N : s.normal) (g : α) :
theorem normal_of_eq_cosets {α : Type u_1} [group α] (s : subgroup α) (h : (g : α), left_coset g s = right_coset s g) :
theorem normal_of_eq_add_cosets {α : Type u_1} [add_group α] (s : add_subgroup α) (h : (g : α), left_add_coset g s = right_add_coset s g) :
theorem normal_iff_eq_cosets {α : Type u_1} [group α] (s : subgroup α) :
theorem left_add_coset_eq_iff {α : Type u_1} [add_group α] (s : add_subgroup α) {x y : α} :
theorem left_coset_eq_iff {α : Type u_1} [group α] (s : subgroup α) {x y : α} :
theorem right_coset_eq_iff {α : Type u_1} [group α] (s : subgroup α) {x y : α} :
theorem right_add_coset_eq_iff {α : Type u_1} [add_group α] (s : add_subgroup α) {x y : α} :
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.

Equations
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
theorem quotient_group.left_rel_apply {α : Type u_1} [group α] {s : subgroup α} {x y : α} :
theorem quotient_add_group.left_rel_apply {α : Type u_1} [add_group α] {s : add_subgroup α} {x y : α} :
setoid.r x y -x + y s
theorem quotient_add_group.left_rel_eq {α : Type u_1} [add_group α] (s : add_subgroup α) :
setoid.r = λ (x y : α), -x + y s
theorem quotient_group.left_rel_eq {α : Type u_1} [group α] (s : subgroup α) :
setoid.r = λ (x y : α), x⁻¹ * y s
@[protected, instance]
def quotient_group.left_rel_decidable {α : Type u_1} [group α] (s : subgroup α) [decidable_pred (λ (_x : α), _x s)] :
Equations
@[protected, instance]
Equations
@[protected, instance]

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

Equations
@[protected, instance]

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

Equations
theorem quotient_group.right_rel_apply {α : Type u_1} [group α] {s : subgroup α} {x y : α} :
theorem quotient_add_group.right_rel_apply {α : Type u_1} [add_group α] {s : add_subgroup α} {x y : α} :
setoid.r x y y + -x s
theorem quotient_add_group.right_rel_eq {α : Type u_1} [add_group α] (s : add_subgroup α) :
setoid.r = λ (x y : α), y + -x s
theorem quotient_group.right_rel_eq {α : Type u_1} [group α] (s : subgroup α) :
setoid.r = λ (x y : α), y * x⁻¹ s
@[protected, instance]
def quotient_group.right_rel_decidable {α : Type u_1} [group α] (s : subgroup α) [decidable_pred (λ (_x : α), _x s)] :
Equations
@[protected, instance]
Equations

Right cosets are in bijection with left cosets.

Equations

Right cosets are in bijection with left cosets.

Equations
@[protected, instance]
def quotient_group.fintype {α : Type u_1} [group α] [fintype α] (s : subgroup α) [decidable_rel setoid.r] :
fintype s)
Equations
@[reducible]
def quotient_add_group.mk {α : Type u_1} [add_group α] {s : add_subgroup α} (a : α) :
α s

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

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

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 : α s Prop} (x : α 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 : α s Prop} (x : α 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 : α s Prop} (x : α s) (H : (z : α), C z) :
C x
theorem quotient_group.induction_on' {α : Type u_1} [group α] {s : subgroup α} {C : α s Prop} (x : α s) (H : (z : α), C z) :
C x
@[simp]
theorem quotient_add_group.quotient_lift_on_coe {α : Type u_1} [add_group α] {s : add_subgroup α} {β : Sort u_2} (f : α β) (h : (a b : α), setoid.r a b f a = f b) (x : α) :
@[simp]
theorem quotient_group.quotient_lift_on_coe {α : Type u_1} [group α] {s : subgroup α} {β : Sort u_2} (f : α β) (h : (a b : α), setoid.r a b f a = f b) (x : α) :
theorem quotient_add_group.forall_coe {α : Type u_1} [add_group α] {s : add_subgroup α} {C : α s Prop} :
( (x : α s), C x) (x : α), C x
theorem quotient_group.forall_coe {α : Type u_1} [group α] {s : subgroup α} {C : α s Prop} :
( (x : α s), C x) (x : α), C x
theorem quotient_add_group.exists_coe {α : Type u_1} [add_group α] {s : add_subgroup α} {C : α s Prop} :
( (x : α s), C x) (x : α), C x
theorem quotient_group.exists_coe {α : Type u_1} [group α] {s : subgroup α} {C : α s Prop} :
( (x : α s), C x) (x : α), C x
@[protected]
theorem quotient_add_group.eq {α : Type u_1} [add_group α] {s : add_subgroup α} {a b : α} :
a = b -a + b s
@[protected]
theorem quotient_group.eq {α : Type u_1} [group α] {s : subgroup α} {a b : α} :
a = b a⁻¹ * b s
theorem quotient_group.eq' {α : Type u_1} [group α] {s : subgroup α} {a b : α} :
@[simp]
theorem quotient_group.out_eq' {α : Type u_1} [group α] {s : subgroup α} (a : α s) :
@[simp]
theorem quotient_add_group.out_eq' {α : Type u_1} [add_group α] {s : add_subgroup α} (a : α s) :
theorem quotient_group.mk_out'_eq_mul {α : Type u_1} [group α] (s : subgroup α) (g : α) :
@[simp]
theorem quotient_add_group.mk_add_of_mem {α : Type u_1} [add_group α] {s : add_subgroup α} {b : α} (a : α) (hb : b s) :
@[simp]
theorem quotient_group.mk_mul_of_mem {α : Type u_1} [group α] {s : subgroup α} {b : α} (a : α) (hb : b s) :
theorem quotient_group.eq_class_eq_left_coset {α : Type u_1} [group α] (s : subgroup α) (g : α) :
{x : α | x = g} = left_coset g 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} = left_add_coset g 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

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

Equations
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

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

Equations
noncomputable def subgroup.group_equiv_quotient_times_subgroup {α : Type u_1} [group α] {s : subgroup α} :
α s) × s

A (non-canonical) bijection between a group α and the product (α/s) × s

Equations
noncomputable def add_subgroup.add_group_equiv_quotient_times_add_subgroup {α : Type u_1} [add_group α] {s : add_subgroup α} :
α s) × s

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

Equations
def add_subgroup.quotient_equiv_of_eq {α : Type u_1} [add_group α] {s t : add_subgroup α} (h : s = t) :
α s α t

If two subgroups M and N of G are equal, their quotients are in bijection.

Equations
def subgroup.quotient_equiv_of_eq {α : Type u_1} [group α] {s t : subgroup α} (h : s = t) :
α s α t

If two subgroups M and N of G are equal, their quotients are in bijection.

Equations
@[simp]
theorem subgroup.quotient_equiv_prod_of_le'_symm_apply {α : Type u_1} [group α] {s t : subgroup α} (h_le : s t) (f : α t α) (hf : function.right_inverse f quotient_group.mk) (a : t) × t s.subgroup_of t) :
((subgroup.quotient_equiv_prod_of_le' h_le f hf).symm) a = quotient.map' (λ (b : t), f a.fst * b) _ a.snd
def subgroup.quotient_equiv_prod_of_le' {α : Type u_1} [group α] {s t : subgroup α} (h_le : s t) (f : α t α) (hf : function.right_inverse f quotient_group.mk) :
α s t) × t s.subgroup_of t

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
@[simp]
theorem add_subgroup.quotient_equiv_sum_of_le'_symm_apply {α : Type u_1} [add_group α] {s t : add_subgroup α} (h_le : s t) (f : α t α) (hf : function.right_inverse f quotient_add_group.mk) (a : t) × t s.add_subgroup_of t) :
@[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) (f : α t α) (hf : function.right_inverse f quotient_add_group.mk) (a : α s) :
(add_subgroup.quotient_equiv_sum_of_le' h_le f hf) a = (quotient.map' id _ a, quotient.map' (λ (g : α), -f (quotient.mk' g) + g, _⟩) _ a)
def add_subgroup.quotient_equiv_sum_of_le' {α : Type u_1} [add_group α] {s t : add_subgroup α} (h_le : s t) (f : α t α) (hf : function.right_inverse f quotient_add_group.mk) :
α s t) × t s.add_subgroup_of t

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
@[simp]
theorem subgroup.quotient_equiv_prod_of_le'_apply {α : Type u_1} [group α] {s t : subgroup α} (h_le : s t) (f : α t α) (hf : function.right_inverse f quotient_group.mk) (a : α 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)
@[simp]
theorem subgroup.quotient_equiv_prod_of_le_symm_apply {α : Type u_1} [group α] {s t : subgroup α} (h_le : s t) (a : t) × t s.subgroup_of t) :
noncomputable def add_subgroup.quotient_equiv_sum_of_le {α : Type u_1} [add_group α] {s t : add_subgroup α} (h_le : s t) :
α s t) × t s.add_subgroup_of t

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

Equations
@[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 : α s) :
@[simp]
noncomputable def subgroup.quotient_equiv_prod_of_le {α : Type u_1} [group α] {s t : subgroup α} (h_le : s t) :
α s t) × t s.subgroup_of t

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 : α s) :

If s ≤ t, then there is an embedding s ⧸ H.add_subgroup_of s ↪ t ⧸ H.add_subgroup_of t.

Equations
def subgroup.quotient_subgroup_of_embedding_of_le {α : Type u_1} [group α] {s t : subgroup α} (H : subgroup α) (h : s t) :

If s ≤ t, then there is an embedding s ⧸ H.subgroup_of s ↪ t ⧸ H.subgroup_of t.

Equations

If s ≤ t, then there is an map H ⧸ s.add_subgroup_of H → H ⧸ t.add_subgroup_of H.

Equations
def subgroup.quotient_subgroup_of_map_of_le {α : Type u_1} [group α] {s t : subgroup α} (H : subgroup α) (h : s t) :

If s ≤ t, then there is a map H ⧸ s.subgroup_of H → H ⧸ t.subgroup_of H.

Equations
def subgroup.quotient_map_of_le {α : Type u_1} [group α] {s t : subgroup α} (h : s t) :
α s α t

If s ≤ t, then there is a map α ⧸ s → α ⧸ t.

Equations
def add_subgroup.quotient_map_of_le {α : Type u_1} [add_group α] {s t : add_subgroup α} (h : s t) :
α s α t

If s ≤ t, then there is an map α ⧸ s → α ⧸ t.

Equations
@[simp]
@[simp]
theorem subgroup.quotient_infi_subgroup_of_embedding_apply {α : Type u_1} [group α] {ι : Type u_2} (f : ι subgroup α) (H : subgroup α) (q : H ( (i : ι), f i).subgroup_of H) (i : ι) :
def subgroup.quotient_infi_subgroup_of_embedding {α : Type u_1} [group α] {ι : Type u_2} (f : ι subgroup α) (H : subgroup α) :
H ( (i : ι), f i).subgroup_of H Π (i : ι), H (f i).subgroup_of H

The natural embedding H ⧸ (⨅ i, f i).subgroup_of H ↪ Π i, H ⧸ (f i).subgroup_of H.

Equations
def add_subgroup.quotient_infi_add_subgroup_of_embedding {α : Type u_1} [add_group α] {ι : Type u_2} (f : ι add_subgroup α) (H : add_subgroup α) :
H ( (i : ι), f i).add_subgroup_of H Π (i : ι), H (f i).add_subgroup_of H

The natural embedding H ⧸ (⨅ i, f i).add_subgroup_of H) ↪ Π i, H ⧸ (f i).add_subgroup_of H.

Equations
@[simp]
theorem add_subgroup.quotient_infi_embedding_apply {α : Type u_1} [add_group α] {ι : Type u_2} (f : ι add_subgroup α) (q : α (i : ι), f i) (i : ι) :
@[simp]
theorem subgroup.quotient_infi_embedding_apply {α : Type u_1} [group α] {ι : Type u_2} (f : ι subgroup α) (q : α (i : ι), f i) (i : ι) :
def subgroup.quotient_infi_embedding {α : Type u_1} [group α] {ι : Type u_2} (f : ι subgroup α) :
(i : ι), f i) Π (i : ι), α f i

The natural embedding α ⧸ (⨅ i, f i) ↪ Π i, α ⧸ f i.

Equations
def add_subgroup.quotient_infi_embedding {α : Type u_1} [add_group α] {ι : Type u_2} (f : ι add_subgroup α) :
(i : ι), f i) Π (i : ι), α f i

The natural embedding α ⧸ (⨅ i, f i) ↪ Π i, α ⧸ f i.

Equations
@[simp]
theorem subgroup.quotient_infi_embedding_apply_mk {α : Type u_1} [group α] {ι : Type u_2} (f : ι subgroup α) (g : α) (i : ι) :

Lagrange's Theorem: The order of an additive subgroup divides the order of its ambient group.

Lagrange's Theorem: The order of a subgroup divides the order of its ambient group.

theorem subgroup.card_quotient_dvd_card {α : Type u_1} [group α] [fintype α] (s : subgroup α) [decidable_pred (λ (_x : α), _x s)] :
theorem add_subgroup.card_quotient_dvd_card {α : Type u_1} [add_group α] [fintype α] (s : add_subgroup α) [decidable_pred (λ (_x : α), _x s)] :
theorem add_subgroup.card_dvd_of_injective {α : Type u_1} [add_group α] {H : Type u_2} [add_group H] [fintype α] [fintype H] (f : α →+ H) (hf : function.injective f) :
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) :
noncomputable def quotient_group.preimage_mk_equiv_subgroup_times_set {α : Type u_1} [group α] (s : subgroup α) (t : set s)) :

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

If s is a subgroup of the additive 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

We use the class has_coe_t instead of has_coe if the first argument is a variable, or if the second argument is a variable not occurring in the first. Using has_coe would cause looping of type-class inference. See https://leanprover.zulipchat.com/#narrow/stream/113488-general/topic/remove.20all.20instances.20with.20variable.20domain