# mathlibdocumentation

group_theory.specific_groups.alternating

# Alternating Groups #

The alternating group on a finite type α is the subgroup of the permutation group perm α consisting of the even permutations.

## Main definitions #

• alternating_group α is the alternating group on α, defined as a subgroup (perm α).

## Main results #

• two_mul_card_alternating_group shows that the alternating group is half as large as the permutation group it is a subgroup of.

• closure_three_cycles_eq_alternating shows that the alternating group is generated by 3-cycles.

• alternating_group.is_simple_group_five shows that the alternating group on fin 5 is simple. The proof shows that the normal closure of any non-identity element of this group contains a 3-cycle.

## Tags #

alternating group permutation

## TODO #

• Show that alternating_group α is simple if and only if fintype.card α ≠ 4.
@[protected, instance]
def alternating_group.fintype (α : Type u_1) [fintype α] [decidable_eq α] :
def alternating_group (α : Type u_1) [fintype α] [decidable_eq α] :

The alternating group on a finite type, realized as a subgroup of equiv.perm. For $A_n$, use alternating_group (fin n).

Equations
Instances for alternating_group
Instances for ↥alternating_group
@[protected, instance]
def alternating_group.unique (α : Type u_1) [fintype α] [decidable_eq α] [subsingleton α] :
Equations
theorem alternating_group_eq_sign_ker {α : Type u_1} [fintype α] [decidable_eq α] :
@[simp]
theorem equiv.perm.mem_alternating_group {α : Type u_1} [fintype α] [decidable_eq α] {f : equiv.perm α} :
theorem equiv.perm.prod_list_swap_mem_alternating_group_iff_even_length {α : Type u_1} [fintype α] [decidable_eq α] {l : list (equiv.perm α)} (hl : ∀ (g : , g l → g.is_swap) :
theorem two_mul_card_alternating_group {α : Type u_1} [fintype α] [decidable_eq α] [nontrivial α] :
@[protected, instance]
def alternating_group.normal {α : Type u_1} [fintype α] [decidable_eq α] :
theorem alternating_group.is_conj_of {α : Type u_1} [fintype α] [decidable_eq α] {σ τ : } (hc : τ) (hσ : σ.support.card + 2 ) :
τ
theorem alternating_group.is_three_cycle_is_conj {α : Type u_1} [fintype α] [decidable_eq α] (h5 : 5 ) {σ τ : } (hσ : σ.is_three_cycle) (hτ : τ.is_three_cycle) :
τ

A key lemma to prove $A_5$ is simple. Shows that any normal subgroup of an alternating group on at least 5 elements is the entire alternating group if it contains a 3-cycle.

Part of proving $A_5$ is simple. Shows that the square of any element of $A_5$ with a 3-cycle in its cycle decomposition is a 3-cycle, so the normal closure of the original element must be $A_5$.

theorem alternating_group.nontrivial_of_three_le_card {α : Type u_1} [fintype α] [decidable_eq α] (h3 : 3 ) :
@[protected, instance]

The normal closure of the 5-cycle fin_rotate 5 within $A_5$ is the whole group. This will be used to show that the normal closure of any 5-cycle within $A_5$ is the whole group.

The normal closure of $(04)(13)$ within $A_5$ is the whole group. This will be used to show that the normal closure of any permutation of cycle type $(2,2)$ is the whole group.

theorem alternating_group.is_conj_swap_mul_swap_of_cycle_type_two {g : equiv.perm (fin 5)} (ha : g ) (h1 : g 1) (h2 : ∀ (n : ), n g.cycle_typen = 2) :
is_conj 4 * 3) g

Shows that any non-identity element of $A_5$ whose cycle decomposition consists only of swaps is conjugate to $(04)(13)$. This is used to show that the normal closure of such a permutation in $A_5$ is $A_5$.

@[protected, instance]

Shows that $A_5$ is simple by taking an arbitrary non-identity element and showing by casework on its cycle type that its normal closure is all of $A_5$.