mathlib documentation

group_theory.perm.cycles

Cyclic permutations #

Main definitions #

In the following, f : equiv.perm β.

The following two definitions require that β is a fintype:

is_cycle #

def equiv.perm.is_cycle {β : Type u_2} (f : equiv.perm β) :
Prop

A permutation is a cycle when any two nonfixed points of the permutation are related by repeated application of the permutation.

Equations
theorem equiv.perm.is_cycle.ne_one {β : Type u_2} {f : equiv.perm β} (h : f.is_cycle) :
f 1
theorem equiv.perm.is_cycle.two_le_card_support {α : Type u_1} [decidable_eq α] [fintype α] {f : equiv.perm α} (h : f.is_cycle) :
theorem equiv.perm.is_cycle.swap {α : Type u_1} [decidable_eq α] {x y : α} (hxy : x y) :
theorem equiv.perm.is_cycle.inv {β : Type u_2} {f : equiv.perm β} (hf : f.is_cycle) :
theorem equiv.perm.is_cycle.exists_gpow_eq {β : Type u_2} {f : equiv.perm β} (hf : f.is_cycle) {x y : β} (hx : f x x) (hy : f y y) :
∃ (i : ), (f ^ i) x = y
theorem equiv.perm.is_cycle.exists_pow_eq {β : Type u_2} [fintype β] {f : equiv.perm β} (hf : f.is_cycle) {x y : β} (hx : f x x) (hy : f y y) :
∃ (i : ), (f ^ i) x = y

The subgroup generated by a cycle is in bijection with its support

Equations
theorem equiv.perm.order_of_is_cycle {α : Type u_1} [decidable_eq α] [fintype α] {σ : equiv.perm α} (hσ : σ.is_cycle) :
theorem equiv.perm.is_cycle_swap_mul_aux₁ {α : Type u_1} [decidable_eq α] (n : ) {b x : α} {f : equiv.perm α} (hb : ((equiv.swap x (f x)) * f) b b) (h : (f ^ n) (f x) = b) :
∃ (i : ), (((equiv.swap x (f x)) * f) ^ i) (f x) = b
theorem equiv.perm.is_cycle_swap_mul_aux₂ {α : Type u_1} [decidable_eq α] (n : ) {b x : α} {f : equiv.perm α} (hb : ((equiv.swap x (f x)) * f) b b) (h : (f ^ n) (f x) = b) :
∃ (i : ), (((equiv.swap x (f x)) * f) ^ i) (f x) = b
theorem equiv.perm.is_cycle.eq_swap_of_apply_apply_eq_self {α : Type u_1} [decidable_eq α] {f : equiv.perm α} (hf : f.is_cycle) {x : α} (hfx : f x x) (hffx : f (f x) = x) :
f = equiv.swap x (f x)
theorem equiv.perm.is_cycle.swap_mul {α : Type u_1} [decidable_eq α] {f : equiv.perm α} (hf : f.is_cycle) {x : α} (hx : f x x) (hffx : f (f x) x) :
((equiv.swap x (f x)) * f).is_cycle
theorem equiv.perm.is_cycle.sign {α : Type u_1} [decidable_eq α] [fintype α] {f : equiv.perm α} (hf : f.is_cycle) :
theorem equiv.perm.is_cycle_of_is_cycle_pow {α : Type u_1} [decidable_eq α] [fintype α] {σ : equiv.perm α} {n : } (h1 : ^ n).is_cycle) (h2 : σ.support ^ n).support) :

same_cycle #

def equiv.perm.same_cycle {β : Type u_2} (f : equiv.perm β) (x y : β) :
Prop

The equivalence relation indicating that two points are in the same cycle of a permutation.

Equations
theorem equiv.perm.same_cycle.refl {β : Type u_2} (f : equiv.perm β) (x : β) :
theorem equiv.perm.same_cycle.symm {β : Type u_2} (f : equiv.perm β) {x y : β} :
f.same_cycle x yf.same_cycle y x
theorem equiv.perm.same_cycle.trans {β : Type u_2} (f : equiv.perm β) {x y z : β} :
f.same_cycle x yf.same_cycle y zf.same_cycle x z
theorem equiv.perm.same_cycle.apply_eq_self_iff {β : Type u_2} {f : equiv.perm β} {x y : β} :
f.same_cycle x y(f x = x f y = y)
theorem equiv.perm.is_cycle.same_cycle {β : Type u_2} {f : equiv.perm β} (hf : f.is_cycle) {x y : β} (hx : f x x) (hy : f y y) :
@[instance]
Equations
theorem equiv.perm.same_cycle_apply {β : Type u_2} {f : equiv.perm β} {x y : β} :
f.same_cycle x (f y) f.same_cycle x y
theorem equiv.perm.same_cycle_cycle {β : Type u_2} {f : equiv.perm β} {x : β} (hx : f x x) :
f.is_cycle ∀ {y : β}, f.same_cycle x y f y y
theorem equiv.perm.same_cycle_inv {β : Type u_2} (f : equiv.perm β) {x y : β} :
theorem equiv.perm.same_cycle_inv_apply {β : Type u_2} {f : equiv.perm β} {x y : β} :

cycle_of #

def equiv.perm.cycle_of {α : Type u_1} [decidable_eq α] [fintype α] (f : equiv.perm α) (x : α) :

f.cycle_of x is the cycle of the permutation f to which x belongs.

Equations
theorem equiv.perm.cycle_of_apply {α : Type u_1} [decidable_eq α] [fintype α] (f : equiv.perm α) (x y : α) :
(f.cycle_of x) y = ite (f.same_cycle x y) (f y) y
theorem equiv.perm.cycle_of_inv {α : Type u_1} [decidable_eq α] [fintype α] (f : equiv.perm α) (x : α) :
@[simp]
theorem equiv.perm.cycle_of_pow_apply_self {α : Type u_1} [decidable_eq α] [fintype α] (f : equiv.perm α) (x : α) (n : ) :
(f.cycle_of x ^ n) x = (f ^ n) x
@[simp]
theorem equiv.perm.cycle_of_gpow_apply_self {α : Type u_1} [decidable_eq α] [fintype α] (f : equiv.perm α) (x : α) (n : ) :
(f.cycle_of x ^ n) x = (f ^ n) x
theorem equiv.perm.same_cycle.cycle_of_apply {α : Type u_1} [decidable_eq α] [fintype α] {f : equiv.perm α} {x y : α} (h : f.same_cycle x y) :
(f.cycle_of x) y = f y
theorem equiv.perm.cycle_of_apply_of_not_same_cycle {α : Type u_1} [decidable_eq α] [fintype α] {f : equiv.perm α} {x y : α} (h : ¬f.same_cycle x y) :
(f.cycle_of x) y = y
@[simp]
theorem equiv.perm.cycle_of_apply_self {α : Type u_1} [decidable_eq α] [fintype α] (f : equiv.perm α) (x : α) :
(f.cycle_of x) x = f x
theorem equiv.perm.is_cycle.cycle_of_eq {α : Type u_1} [decidable_eq α] [fintype α] {f : equiv.perm α} (hf : f.is_cycle) {x : α} (hx : f x x) :
f.cycle_of x = f
@[simp]
theorem equiv.perm.cycle_of_eq_one_iff {α : Type u_1} [decidable_eq α] [fintype α] (f : equiv.perm α) {x : α} :
f.cycle_of x = 1 f x = x
theorem equiv.perm.is_cycle.cycle_of {α : Type u_1} [decidable_eq α] [fintype α] {f : equiv.perm α} (hf : f.is_cycle) {x : α} :
f.cycle_of x = ite (f x = x) 1 f
theorem equiv.perm.cycle_of_one {α : Type u_1} [decidable_eq α] [fintype α] (x : α) :
1.cycle_of x = 1
theorem equiv.perm.is_cycle_cycle_of {α : Type u_1} [decidable_eq α] [fintype α] (f : equiv.perm α) {x : α} (hx : f x x) :

cycle_factors #

def equiv.perm.cycle_factors_aux {α : Type u_1} [decidable_eq α] [fintype α] (l : list α) (f : equiv.perm α) :
(∀ {x : α}, f x xx l){l // l.prod = f (∀ (g : equiv.perm α), g l → g.is_cycle) list.pairwise equiv.perm.disjoint l}

Given a list l : list α and a permutation f : perm α whose nonfixed points are all in l, recursively factors f into cycles.

Equations
def equiv.perm.cycle_factors {α : Type u_1} [decidable_eq α] [fintype α] [linear_order α] (f : equiv.perm α) :
{l // l.prod = f (∀ (g : equiv.perm α), g l → g.is_cycle) list.pairwise equiv.perm.disjoint l}

Factors a permutation f into a list of disjoint cyclic permutations that multiply to f.

Equations
def equiv.perm.trunc_cycle_factors {α : Type u_1} [decidable_eq α] [fintype α] (f : equiv.perm α) :
trunc {l // l.prod = f (∀ (g : equiv.perm α), g l → g.is_cycle) list.pairwise equiv.perm.disjoint l}

Factors a permutation f into a list of disjoint cyclic permutations that multiply to f, without a linear order.

Equations
theorem equiv.perm.cycle_induction_on {β : Type u_2} [fintype β] {P : equiv.perm β → Prop} (σ : equiv.perm β) (base_one : P 1) (base_cycles : ∀ (σ : equiv.perm β), σ.is_cycleP σ) (induction_disjoint : ∀ (σ τ : equiv.perm β), σ.disjoint τP σP τP * τ)) :
P σ

Fixed points #

theorem equiv.perm.fixed_point_card_lt_of_ne_one {α : Type u_1} [decidable_eq α] [fintype α] {σ : equiv.perm α} (h : σ 1) :
(finset.filter (λ (x : α), σ x = x) finset.univ).card < fintype.card α - 1