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.

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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

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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
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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
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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

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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
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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
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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
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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
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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.

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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.

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theorem left_add_coset_equivalence_rel {α : Type u_1} [has_add α] (s : set α) :
equivalence (left_add_coset_equivalence s)
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theorem left_coset_equivalence_rel {α : Type u_1} [has_mul α] (s : set α) :
equivalence (left_coset_equivalence s)
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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
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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.

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theorem right_add_coset_equivalence_rel {α : Type u_1} [has_add α] (s : set α) :
equivalence (right_add_coset_equivalence s)
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theorem right_coset_equivalence_rel {α : Type u_1} [has_mul α] (s : set α) :
equivalence (right_coset_equivalence s)
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@[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
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@[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
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@[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
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@[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)
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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)
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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)
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@[simp]
theorem one_left_coset {α : Type u_1} [monoid α] (s : set α) :
1 *l s = s
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@[simp]
theorem zero_left_add_coset {α : Type u_1} [add_monoid α] (s : set α) :
0 +l s = s
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@[simp]
theorem right_coset_one {α : Type u_1} [monoid α] (s : set α) :
s *r 1 = s
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@[simp]
theorem right_add_coset_zero {α : Type u_1} [add_monoid α] (s : set α) :
s +r 0 = s
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theorem mem_own_left_coset {α : Type u_1} [monoid α] (s : submonoid α) (a : α) :
a ∈ a *l ↑s
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theorem mem_own_left_add_coset {α : Type u_1} [add_monoid α] (s : add_submonoid α) (a : α) :
a ∈ a +l ↑s
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theorem mem_own_right_coset {α : Type u_1} [monoid α] (s : submonoid α) (a : α) :
a ∈ ↑s *r a
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theorem mem_own_right_add_coset {α : Type u_1} [add_monoid α] (s : add_submonoid α) (a : α) :
a ∈ ↑s +r a
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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
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theorem mem_left_coset_left_coset {α : Type u_1} [monoid α] (s : submonoid α) {a : α} (ha : a *l ↑s = ↑s) :
a ∈ s
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theorem mem_right_coset_right_coset {α : Type u_1} [monoid α] (s : submonoid α) {a : α} (ha : ↑s *r a = ↑s) :
a ∈ s
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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
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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
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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
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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
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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
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theorem left_coset_mem_left_coset {α : Type u_1} [group α] (s : subgroup α) {a : α} (ha : a ∈ s) :
a *l ↑s = ↑s
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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
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theorem right_coset_mem_right_coset {α : Type u_1} [group α] (s : subgroup α) {a : α} (ha : a ∈ s) :
↑s *r a = ↑s
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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
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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
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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
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theorem normal_of_eq_cosets {α : Type u_1} [group α] (s : subgroup α) (h : ∀ (g : α), g *l ↑s = ↑s *r g) :
s.normal
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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) :
s.normal
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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
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theorem normal_iff_eq_cosets {α : Type u_1} [group α] (s : subgroup α) :
s.normal ↔ ∀ (g : α), g *l ↑s = ↑s *r g
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def quotient_add_group.left_rel {α : Type u_1} [add_group α] (s : add_subgroup α) :
setoid α

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

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

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

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

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

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

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

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

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@[instance]
def quotient_group.right_rel_decidable {α : Type u_1} [group α] (s : subgroup α) [d : decidable_pred (λ (_x : α), _x ∈ s)] :
decidable_rel setoid.r
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@[instance]
def quotient_add_group.right_rel_decidable {α : Type u_1} [add_group α] (s : add_subgroup α) [d : decidable_pred (λ (_x : α), _x ∈ s)] :
decidable_rel setoid.r
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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

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

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

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

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

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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
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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
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@[instance]
def quotient_group.quotient.has_coe_t {α : Type u_1} [group α] {s : subgroup α} :
has_coe_t α (quotient_group.quotient s)
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@[instance]
def quotient_add_group.quotient.has_coe_t {α : Type u_1} [add_group α] {s : add_subgroup α} :
has_coe_t α (quotient_add_group.quotient s)
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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
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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
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@[instance]
def quotient_group.quotient.inhabited {α : Type u_1} [group α] (s : subgroup α) :
inhabited (quotient_group.quotient s)
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@[instance]
def quotient_add_group.quotient.inhabited {α : Type u_1} [add_group α] (s : add_subgroup α) :
inhabited (quotient_add_group.quotient s)
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theorem quotient_add_group.eq {α : Type u_1} [add_group α] {s : add_subgroup α} {a b : α} :
↑a = ↑b ↔ -a + b ∈ s
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theorem quotient_group.eq {α : Type u_1} [group α] {s : subgroup α} {a b : α} :
↑a = ↑b ↔ a⁻¹ * b ∈ s
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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
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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
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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
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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
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def subgroup.left_coset_equiv_subgroup {α : Type u_1} [group α] {s : subgroup α} (g : α) :
↥(g *l ↑s) ≃ ↥s

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

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def add_subgroup.left_coset_equiv_add_subgroup {α : Type u_1} [add_group α] {s : add_subgroup α} (g : α) :
↥(g +l ↑s) ≃ ↥s

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

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def subgroup.right_coset_equiv_subgroup {α : Type u_1} [group α] {s : subgroup α} (g : α) :
↥(↑s *r g) ≃ ↥s

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

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def add_subgroup.right_coset_equiv_add_subgroup {α : Type u_1} [add_group α] {s : add_subgroup α} (g : α) :
↥(↑s +r g) ≃ ↥s

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

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def subgroup.group_equiv_quotient_times_subgroup {α : Type u_1} [group α] {s : subgroup α} :
α ≃ quotient_group.quotient s × ↥s

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

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def add_subgroup.add_group_equiv_quotient_times_add_subgroup {α : Type u_1} [add_group α] {s : add_subgroup α} :
α ≃ quotient_add_group.quotient s × ↥s

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

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theorem subgroup.card_eq_card_quotient_mul_card_subgroup {α : Type u_1} [group α] [fintype α] (s : subgroup α) [fintype ↥s] [decidable_pred (λ (a : α), a ∈ s)] :
fintype.card α = (fintype.card (quotient_group.quotient s)) * fintype.card ↥s
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theorem subgroup.card_subgroup_dvd_card {α : Type u_1} [group α] [fintype α] (s : subgroup α) [fintype ↥s] :
fintype.card ↥s ∣ fintype.card α

Order of a Subgroup

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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] :
fintype.card (quotient_group.quotient s) ∣ fintype.card α
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def quotient_group.preimage_mk_equiv_subgroup_times_set {α : Type u_1} [group α] (s : subgroup α) (t : set (quotient_group.quotient s)) :
↥(quotient_group.mk ⁻¹' t) ≃ ↥s × ↥t

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.

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