mathlib documentation

group_theory.complement

Complements #

In this file we define the complement of a subgroup.

Main definitions #

Main results #

def add_subgroup.is_complement {G : Type u_1} [add_group G] (S T : set G) :
Prop

S and T are complements if (*) : S × T → G is a bijection

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

S and T are complements if (*) : S × T → G is a bijection. This notion generalizes left transversals, right transversals, and complementary subgroups.

Equations
@[reducible]
def add_subgroup.is_complement' {G : Type u_1} [add_group G] (H K : add_subgroup G) :
Prop

H and K are complements if (*) : H × K → G is a bijection

@[reducible]
def subgroup.is_complement' {G : Type u_1} [group G] (H K : subgroup G) :
Prop

H and K are complements if (*) : H × K → G is a bijection

def add_subgroup.left_transversals {G : Type u_1} [add_group G] (T : set G) :
set (set G)

The set of left-complements of T : set G

Equations
Instances for add_subgroup.left_transversals
def subgroup.left_transversals {G : Type u_1} [group G] (T : set G) :
set (set G)

The set of left-complements of T : set G

Equations
Instances for subgroup.left_transversals
def subgroup.right_transversals {G : Type u_1} [group G] (S : set G) :
set (set G)

The set of right-complements of S : set G

Equations
Instances for subgroup.right_transversals
def add_subgroup.right_transversals {G : Type u_1} [add_group G] (S : set G) :
set (set G)

The set of right-complements of S : set G

Equations
Instances for add_subgroup.right_transversals
theorem subgroup.is_complement_iff_exists_unique {G : Type u_1} [group G] {S T : set G} :
subgroup.is_complement S T (g : G), ∃! (x : S × T), x.fst.val * x.snd.val = g
theorem subgroup.is_complement.exists_unique {G : Type u_1} [group G] {S T : set G} (h : subgroup.is_complement S T) (g : G) :
∃! (x : S × T), x.fst.val * x.snd.val = g
theorem add_subgroup.is_complement.exists_unique {G : Type u_1} [add_group G] {S T : set G} (h : add_subgroup.is_complement S T) (g : G) :
∃! (x : S × T), x.fst.val + x.snd.val = g
theorem subgroup.is_complement'.symm {G : Type u_1} [group G] {H K : subgroup G} (h : H.is_complement' K) :
theorem subgroup.is_complement_singleton_left {G : Type u_1} [group G] {S : set G} {g : G} :
theorem subgroup.is_complement_singleton_right {G : Type u_1} [group G] {S : set G} {g : G} :
theorem subgroup.is_complement_top_left {G : Type u_1} [group G] {S : set G} :
theorem subgroup.is_complement_top_right {G : Type u_1} [group G] {S : set G} :
@[simp]
@[simp]
@[simp]
@[simp]
theorem subgroup.range_mem_left_transversals {G : Type u_1} [group G] {H : subgroup G} {f : G H G} (hf : (q : G H), (f q) = q) :
theorem add_subgroup.range_mem_left_transversals {G : Type u_1} [add_group G] {H : add_subgroup G} {f : G H G} (hf : (q : G H), (f q) = q) :
theorem subgroup.exists_left_transversal {G : Type u_1} [group G] {H : subgroup G} (g : G) :
theorem subgroup.exists_right_transversal {G : Type u_1} [group G] {H : subgroup G} (g : G) :
noncomputable def subgroup.mem_left_transversals.to_equiv {G : Type u_1} [group G] {H : subgroup G} {S : set G} (hS : S subgroup.left_transversals H) :
G H S

A left transversal is in bijection with left cosets.

Equations
noncomputable def add_subgroup.mem_left_transversals.to_equiv {G : Type u_1} [add_group G] {H : add_subgroup G} {S : set G} (hS : S add_subgroup.left_transversals H) :
G H S

A left transversal is in bijection with left cosets.

Equations
theorem add_subgroup.mem_left_transversals.to_equiv_apply {G : Type u_1} [add_group G] {H : add_subgroup G} {f : G H G} (hf : (q : G H), (f q) = q) (q : G H) :
theorem subgroup.mem_left_transversals.to_equiv_apply {G : Type u_1} [group G] {H : subgroup G} {f : G H G} (hf : (q : G H), (f q) = q) (q : G H) :
noncomputable def subgroup.mem_left_transversals.to_fun {G : Type u_1} [group G] {H : subgroup G} {S : set G} (hS : S subgroup.left_transversals H) :
G S

A left transversal can be viewed as a function mapping each element of the group to the chosen representative from that left coset.

Equations
noncomputable def add_subgroup.mem_left_transversals.to_fun {G : Type u_1} [add_group G] {H : add_subgroup G} {S : set G} (hS : S add_subgroup.left_transversals H) :
G S

A left transversal can be viewed as a function mapping each element of the group to the chosen representative from that left coset.

Equations

A right transversal is in bijection with right cosets.

Equations

A right transversal can be viewed as a function mapping each element of the group to the chosen representative from that right coset.

Equations
noncomputable def subgroup.mem_right_transversals.to_fun {G : Type u_1} [group G] {H : subgroup G} {S : set G} (hS : S subgroup.right_transversals H) :
G S

A right transversal can be viewed as a function mapping each element of the group to the chosen representative from that right coset.

Equations
theorem subgroup.is_complement'.is_compl {G : Type u_1} [group G] {H K : subgroup G} (h : H.is_complement' K) :
theorem subgroup.is_complement'.sup_eq_top {G : Type u_1} [group G] {H K : subgroup G} (h : H.is_complement' K) :
H K =
theorem subgroup.is_complement'.disjoint {G : Type u_1} [group G] {H K : subgroup G} (h : H.is_complement' K) :
theorem subgroup.is_complement'_stabilizer {G : Type u_1} [group G] {H : subgroup G} {α : Type u_2} [mul_action G α] (a : α) (h1 : (h : H), h a = a h = 1) (h2 : (g : G), (h : H), h g a = a) :
noncomputable def subgroup.transfer_function {G : Type u} [group G] (H : subgroup G) (g : G) :
G H G

The transfer transversal as a function. Given a ⟨g⟩-orbit q₀, g • q₀, ..., g ^ (m - 1) • q₀ in G ⧸ H, an element g ^ k • q₀ is mapped to g ^ k • g₀ for a fixed choice of representative g₀ of q₀.

Equations
theorem subgroup.coe_transfer_function {G : Type u} [group G] (H : subgroup G) (g : G) (q : G H) :
def subgroup.transfer_set {G : Type u} [group G] (H : subgroup G) (g : G) :
set G

The transfer transversal as a set. Contains elements of the form g ^ k • g₀ for fixed choices of representatives g₀ of fixed choices of representatives q₀ of ⟨g⟩-orbits in G ⧸ H.

Equations
theorem subgroup.mem_transfer_set {G : Type u} [group G] (H : subgroup G) (g : G) (q : G H) :

The transfer transversal. Contains elements of the form g ^ k • g₀ for fixed choices of representatives g₀ of fixed choices of representatives q₀ of ⟨g⟩-orbits in G ⧸ H.

Equations