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:

Main results #

is_cycle #

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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
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theorem equiv.perm.is_cycle.ne_one {β : Type u_2} {f : equiv.perm β} (h : f.is_cycle) :
f ≠ 1
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@[simp]
theorem equiv.perm.not_is_cycle_one {β : Type u_2} :
¬1.is_cycle
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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) :
2 ≤ f.support.card
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theorem equiv.perm.is_cycle_swap {α : Type u_1} [decidable_eq α] {x y : α} (hxy : x ≠ y) :
(equiv.swap x y).is_cycle
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theorem equiv.perm.is_swap.is_cycle {α : Type u_1} [decidable_eq α] {f : equiv.perm α} (hf : f.is_swap) :
f.is_cycle
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theorem equiv.perm.is_cycle.inv {β : Type u_2} {f : equiv.perm β} (hf : f.is_cycle) :
f⁻¹.is_cycle
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theorem equiv.perm.is_cycle.is_cycle_conj {β : Type u_2} {f g : equiv.perm β} (hf : f.is_cycle) :
((g * f) * g⁻¹).is_cycle
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theorem equiv.perm.is_cycle.exists_zpow_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 : ℤ), ⇑(f ^ i) x = y
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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 : ℕ), ⇑(f ^ i) x = y
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theorem equiv.perm.is_cycle.exists_pow_eq_one {β : Type u_2} [fintype β] {f : equiv.perm β} (hf : f.is_cycle) :
∃ (k : ℕ) (hk : 1 < k), f ^ k = 1
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noncomputable def equiv.perm.is_cycle.zpowers_equiv_support {α : Type u_1} [decidable_eq α] [fintype α] {σ : equiv.perm α} (hσ : σ.is_cycle) :
↥↑(subgroup.zpowers σ) ≃ ↥↑(σ.support)

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

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@[simp]
theorem equiv.perm.is_cycle.zpowers_equiv_support_apply {α : Type u_1} [decidable_eq α] [fintype α] {σ : equiv.perm α} (hσ : σ.is_cycle) {n : ℕ} :
⇑(hσ.zpowers_equiv_support) ⟨σ ^ n, _⟩ = ⟨⇑(σ ^ n) (classical.some hσ), _⟩
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@[simp]
theorem equiv.perm.is_cycle.zpowers_equiv_support_symm_apply {α : Type u_1} [decidable_eq α] [fintype α] {σ : equiv.perm α} (hσ : σ.is_cycle) (n : ℕ) :
⇑(hσ.zpowers_equiv_support.symm) ⟨⇑(σ ^ n) (classical.some hσ), _⟩ = ⟨σ ^ n, _⟩
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theorem equiv.perm.order_of_is_cycle {α : Type u_1} [decidable_eq α] [fintype α] {σ : equiv.perm α} (hσ : σ.is_cycle) :
order_of σ = σ.support.card
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theorem equiv.perm.is_cycle_swap_mul_aux₁ {α : Type u_1} [decidable_eq α] (n : ℕ) {b x : α} {f : equiv.perm α} (hb : (⇑(equiv.swap x (⇑f x)) * f) b ≠ b) (h : ⇑(f ^ n) (⇑f x) = b) :
∃ (i : ℤ), ⇑(((equiv.swap x (⇑f x)) * f) ^ i) (⇑f x) = b
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theorem equiv.perm.is_cycle_swap_mul_aux₂ {α : Type u_1} [decidable_eq α] (n : ℤ) {b x : α} {f : equiv.perm α} (hb : (⇑(equiv.swap x (⇑f x)) * f) b ≠ b) (h : ⇑(f ^ n) (⇑f x) = b) :
∃ (i : ℤ), ⇑(((equiv.swap x (⇑f x)) * f) ^ i) (⇑f x) = b
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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)
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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
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theorem equiv.perm.is_cycle.sign {α : Type u_1} [decidable_eq α] [fintype α] {f : equiv.perm α} (hf : f.is_cycle) :
⇑equiv.perm.sign f = -(-1) ^ f.support.card
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theorem equiv.perm.is_cycle_of_is_cycle_pow {α : Type u_1} [decidable_eq α] [fintype α] {σ : equiv.perm α} {n : ℕ} (h1 : (σ ^ n).is_cycle) (h2 : σ.support ≤ (σ ^ n).support) :
σ.is_cycle
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theorem equiv.perm.is_cycle_of_is_cycle_zpow {α : Type u_1} [decidable_eq α] [fintype α] {σ : equiv.perm α} {n : ℤ} (h1 : (σ ^ n).is_cycle) (h2 : σ.support ≤ (σ ^ n).support) :
σ.is_cycle
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theorem equiv.perm.is_cycle.extend_domain {β : Type u_2} {α : Type u_1} {p : β → Prop} [decidable_pred p] (f : α ≃ subtype p) {g : equiv.perm α} (h : g.is_cycle) :
(g.extend_domain f).is_cycle
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theorem equiv.perm.nodup_of_pairwise_disjoint_cycles {β : Type u_2} {l : list (equiv.perm β)} (h1 : ∀ (f : equiv.perm β), f ∈ l → f.is_cycle) (h2 : list.pairwise equiv.perm.disjoint l) :
l.nodup

same_cycle #

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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
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theorem equiv.perm.same_cycle.refl {β : Type u_2} (f : equiv.perm β) (x : β) :
f.same_cycle x x
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theorem equiv.perm.same_cycle.symm {β : Type u_2} {f : equiv.perm β} {x y : β} :
f.same_cycle x y → f.same_cycle y x
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theorem equiv.perm.same_cycle.trans {β : Type u_2} {f : equiv.perm β} {x y z : β} :
f.same_cycle x y → f.same_cycle y z → f.same_cycle x z
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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)
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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) :
f.same_cycle x y
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theorem equiv.perm.same_cycle.nat' {β : Type u_2} [fintype β] {f : equiv.perm β} {x y : β} (h : f.same_cycle x y) :
∃ (i : ℕ) (h : i < order_of f), ⇑(f ^ i) x = y
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theorem equiv.perm.same_cycle.nat'' {β : Type u_2} [fintype β] {f : equiv.perm β} {x y : β} (h : f.same_cycle x y) :
∃ (i : ℕ) (hpos : 0 < i) (h : i ≤ order_of f), ⇑(f ^ i) x = y
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@[protected, instance]
def equiv.perm.same_cycle.decidable_rel {α : Type u_1} [decidable_eq α] [fintype α] (f : equiv.perm α) :
decidable_rel f.same_cycle
Equations
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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
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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
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theorem equiv.perm.same_cycle_inv {β : Type u_2} (f : equiv.perm β) {x y : β} :
f⁻¹.same_cycle x y ↔ f.same_cycle x y
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theorem equiv.perm.same_cycle_inv_apply {β : Type u_2} {f : equiv.perm β} {x y : β} :
f.same_cycle x (⇑f⁻¹ y) ↔ f.same_cycle x y
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@[simp]
theorem equiv.perm.same_cycle_pow_left_iff {β : Type u_2} {f : equiv.perm β} {x y : β} {n : ℕ} :
f.same_cycle (⇑(f ^ n) x) y ↔ f.same_cycle x y
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@[simp]
theorem equiv.perm.same_cycle_zpow_left_iff {β : Type u_2} {f : equiv.perm β} {x y : β} {n : ℤ} :
f.same_cycle (⇑(f ^ n) x) y ↔ f.same_cycle x y
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theorem equiv.perm.is_cycle.support_congr {α : Type u_1} [decidable_eq α] [fintype α] {f g : equiv.perm α} (hf : f.is_cycle) (hg : g.is_cycle) (h : f.support ⊆ g.support) (h' : ∀ (x : α), x ∈ f.support → ⇑f x = ⇑g x) :
f = g

Unlike support_congr, which assumes that ∀ (x ∈ g.support), f x = g x), here we have the weaker assumption that ∀ (x ∈ f.support), f x = g x.

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theorem equiv.perm.is_cycle.eq_on_support_inter_nonempty_congr {α : Type u_1} [decidable_eq α] [fintype α] {f g : equiv.perm α} (hf : f.is_cycle) (hg : g.is_cycle) (h : ∀ (x : α), x ∈ f.support ∩ g.support → ⇑f x = ⇑g x) {x : α} (hx : ⇑f x = ⇑g x) (hx' : x ∈ f.support) :
f = g

If two cyclic permutations agree on all terms in their intersection, and that intersection is not empty, then the two cyclic permutations must be equal.

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theorem equiv.perm.is_cycle.support_pow_eq_iff {α : Type u_1} [decidable_eq α] [fintype α] {f : equiv.perm α} (hf : f.is_cycle) {n : ℕ} :
(f ^ n).support = f.support ↔ ¬order_of f ∣ n
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theorem equiv.perm.is_cycle.pow_iff {β : Type u_2} [fintype β] {f : equiv.perm β} (hf : f.is_cycle) {n : ℕ} :
(f ^ n).is_cycle ↔ n.coprime (order_of f)
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theorem equiv.perm.is_cycle.pow_eq_one_iff {α : Type u_1} [decidable_eq α] [fintype α] {f : equiv.perm α} (hf : f.is_cycle) {n : ℕ} :
f ^ n = 1 ↔ ∃ (x : α) (H : x ∈ f.support), ⇑(f ^ n) x = x
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theorem equiv.perm.is_cycle.mem_support_pos_pow_iff_of_lt_order_of {α : Type u_1} [decidable_eq α] [fintype α] {f : equiv.perm α} (hf : f.is_cycle) {n : ℕ} (npos : 0 < n) (hn : n < order_of f) {x : α} :
x ∈ (f ^ n).support ↔ x ∈ f.support
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theorem equiv.perm.is_cycle.is_cycle_pow_pos_of_lt_prime_order {β : Type u_2} [fintype β] {f : equiv.perm β} (hf : f.is_cycle) (hf' : nat.prime (order_of f)) (n : ℕ) (hn : 0 < n) (hn' : n < order_of f) :
(f ^ n).is_cycle

cycle_of #

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

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

Equations
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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
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theorem equiv.perm.cycle_of_inv {α : Type u_1} [decidable_eq α] [fintype α] (f : equiv.perm α) (x : α) :
(f.cycle_of x)⁻¹ = f⁻¹.cycle_of x
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@[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 : ℕ) :
⇑(f.cycle_of x ^ n) x = ⇑(f ^ n) x
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@[simp]
theorem equiv.perm.cycle_of_zpow_apply_self {α : Type u_1} [decidable_eq α] [fintype α] (f : equiv.perm α) (x : α) (n : ℤ) :
⇑(f.cycle_of x ^ n) x = ⇑(f ^ n) x
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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
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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
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theorem equiv.perm.same_cycle.cycle_of_eq {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 = f.cycle_of y
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@[simp]
theorem equiv.perm.cycle_of_apply_apply_zpow_self {α : Type u_1} [decidable_eq α] [fintype α] (f : equiv.perm α) (x : α) (k : ℤ) :
⇑(f.cycle_of x) (⇑(f ^ k) x) = ⇑(f ^ (k + 1)) x
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@[simp]
theorem equiv.perm.cycle_of_apply_apply_pow_self {α : Type u_1} [decidable_eq α] [fintype α] (f : equiv.perm α) (x : α) (k : ℕ) :
⇑(f.cycle_of x) (⇑(f ^ k) x) = ⇑(f ^ (k + 1)) x
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@[simp]
theorem equiv.perm.cycle_of_apply_apply_self {α : Type u_1} [decidable_eq α] [fintype α] (f : equiv.perm α) (x : α) :
⇑(f.cycle_of x) (⇑f x) = ⇑f (⇑f x)
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@[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
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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
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@[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
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@[simp]
theorem equiv.perm.cycle_of_self_apply {α : Type u_1} [decidable_eq α] [fintype α] (f : equiv.perm α) (x : α) :
f.cycle_of (⇑f x) = f.cycle_of x
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@[simp]
theorem equiv.perm.cycle_of_self_apply_pow {α : Type u_1} [decidable_eq α] [fintype α] (f : equiv.perm α) (n : ℕ) (x : α) :
f.cycle_of (⇑(f ^ n) x) = f.cycle_of x
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@[simp]
theorem equiv.perm.cycle_of_self_apply_zpow {α : Type u_1} [decidable_eq α] [fintype α] (f : equiv.perm α) (n : ℤ) (x : α) :
f.cycle_of (⇑(f ^ n) x) = f.cycle_of x
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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
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theorem equiv.perm.cycle_of_one {α : Type u_1} [decidable_eq α] [fintype α] (x : α) :
1.cycle_of x = 1
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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) :
(f.cycle_of x).is_cycle
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@[simp]
theorem equiv.perm.two_le_card_support_cycle_of_iff {α : Type u_1} [decidable_eq α] [fintype α] {f : equiv.perm α} {x : α} :
2 ≤ (f.cycle_of x).support.card ↔ ⇑f x ≠ x
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@[simp]
theorem equiv.perm.card_support_cycle_of_pos_iff {α : Type u_1} [decidable_eq α] [fintype α] {f : equiv.perm α} {x : α} :
0 < (f.cycle_of x).support.card ↔ ⇑f x ≠ x
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theorem equiv.perm.pow_apply_eq_pow_mod_order_of_cycle_of_apply {α : Type u_1} [decidable_eq α] [fintype α] (f : equiv.perm α) (n : ℕ) (x : α) :
⇑(f ^ n) x = ⇑(f ^ (n % order_of (f.cycle_of x))) x
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theorem equiv.perm.cycle_of_mul_of_apply_right_eq_self {α : Type u_1} [decidable_eq α] [fintype α] {f g : equiv.perm α} (h : commute f g) (x : α) (hx : ⇑g x = x) :
(f * g).cycle_of x = f.cycle_of x
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theorem equiv.perm.disjoint.cycle_of_mul_distrib {α : Type u_1} [decidable_eq α] [fintype α] {f g : equiv.perm α} (h : f.disjoint g) (x : α) :
(f * g).cycle_of x = (f.cycle_of x) * g.cycle_of x
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theorem equiv.perm.support_cycle_of_eq_nil_iff {α : Type u_1} [decidable_eq α] [fintype α] {f : equiv.perm α} {x : α} :
(f.cycle_of x).support = ∅ ↔ x ∉ f.support
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theorem equiv.perm.support_cycle_of_le {α : Type u_1} [decidable_eq α] [fintype α] (f : equiv.perm α) (x : α) :
(f.cycle_of x).support ≤ f.support
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theorem equiv.perm.mem_support_cycle_of_iff {α : Type u_1} [decidable_eq α] [fintype α] {f : equiv.perm α} {x y : α} :
y ∈ (f.cycle_of x).support ↔ f.same_cycle x y ∧ x ∈ f.support
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theorem equiv.perm.same_cycle.mem_support_iff {α : Type u_1} [decidable_eq α] [fintype α] {f : equiv.perm α} {x y : α} (h : f.same_cycle x y) :
x ∈ f.support ↔ y ∈ f.support
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theorem equiv.perm.pow_mod_card_support_cycle_of_self_apply {α : Type u_1} [decidable_eq α] [fintype α] (f : equiv.perm α) (n : ℕ) (x : α) :
⇑(f ^ (n % (f.cycle_of x).support.card)) x = ⇑(f ^ n) x
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theorem equiv.perm.is_cycle_cycle_of_iff {α : Type u_1} [decidable_eq α] [fintype α] (f : equiv.perm α) {x : α} :
(f.cycle_of x).is_cycle ↔ ⇑f x ≠ x

x is in the support of f iff cycle_of f x is a cycle.

cycle_factors #

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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 ≠ x → x ∈ 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
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theorem equiv.perm.mem_list_cycles_iff {α : Type u_1} [fintype α] {l : list (equiv.perm α)} (h1 : ∀ (σ : equiv.perm α), σ ∈ l → σ.is_cycle) (h2 : list.pairwise equiv.perm.disjoint l) {σ : equiv.perm α} :
σ ∈ l ↔ σ.is_cycle ∧ ∀ (a : α), ⇑σ a ≠ a → ⇑σ a = ⇑(l.prod) a
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theorem equiv.perm.list_cycles_perm_list_cycles {α : Type u_1} [fintype α] {l₁ l₂ : list (equiv.perm α)} (h₀ : l₁.prod = l₂.prod) (h₁l₁ : ∀ (σ : equiv.perm α), σ ∈ l₁ → σ.is_cycle) (h₁l₂ : ∀ (σ : equiv.perm α), σ ∈ l₂ → σ.is_cycle) (h₂l₁ : list.pairwise equiv.perm.disjoint l₁) (h₂l₂ : list.pairwise equiv.perm.disjoint l₂) :
l₁ ~ l₂
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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.

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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.

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

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

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theorem equiv.perm.cycle_factors_finset_eq_list_to_finset {α : Type u_1} [decidable_eq α] [fintype α] {σ : equiv.perm α} {l : list (equiv.perm α)} (hn : l.nodup) :
σ.cycle_factors_finset = l.to_finset ↔ (∀ (f : equiv.perm α), f ∈ l → f.is_cycle) ∧ list.pairwise equiv.perm.disjoint l ∧ l.prod = σ
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theorem equiv.perm.cycle_factors_finset_eq_finset {α : Type u_1} [decidable_eq α] [fintype α] {σ : equiv.perm α} {s : finset (equiv.perm α)} :
σ.cycle_factors_finset = s ↔ (∀ (f : equiv.perm α), f ∈ s → f.is_cycle) ∧ ∃ (h : ∀ (a : equiv.perm α), a ∈ s → ∀ (b : equiv.perm α), b ∈ s → a ≠ b → a.disjoint b), s.noncomm_prod id _ = σ
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theorem equiv.perm.cycle_factors_finset_pairwise_disjoint {α : Type u_1} [decidable_eq α] [fintype α] (f p : equiv.perm α) (hp : p ∈ f.cycle_factors_finset) (q : equiv.perm α) (hq : q ∈ f.cycle_factors_finset) (h : p ≠ q) :
p.disjoint q
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theorem equiv.perm.cycle_factors_finset_mem_commute {α : Type u_1} [decidable_eq α] [fintype α] (f p : equiv.perm α) (hp : p ∈ f.cycle_factors_finset) (q : equiv.perm α) (hq : q ∈ f.cycle_factors_finset) :
commute p q
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theorem equiv.perm.cycle_factors_finset_noncomm_prod {α : Type u_1} [decidable_eq α] [fintype α] (f : equiv.perm α) (comm : (∀ (g : equiv.perm α), g ∈ f.cycle_factors_finset → ∀ (h : equiv.perm α), h ∈ f.cycle_factors_finset → commute (id g) (id h)) := _) :
f.cycle_factors_finset.noncomm_prod id comm = f

The product of cycle factors is equal to the original f : perm α.

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theorem equiv.perm.mem_cycle_factors_finset_iff {α : Type u_1} [decidable_eq α] [fintype α] {f p : equiv.perm α} :
p ∈ f.cycle_factors_finset ↔ p.is_cycle ∧ ∀ (a : α), a ∈ p.support → ⇑p a = ⇑f a
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theorem equiv.perm.cycle_of_mem_cycle_factors_finset_iff {α : Type u_1} [decidable_eq α] [fintype α] {f : equiv.perm α} {x : α} :
f.cycle_of x ∈ f.cycle_factors_finset ↔ x ∈ f.support
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theorem equiv.perm.mem_cycle_factors_finset_support_le {α : Type u_1} [decidable_eq α] [fintype α] {p f : equiv.perm α} (h : p ∈ f.cycle_factors_finset) :
p.support ≤ f.support
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theorem equiv.perm.cycle_factors_finset_eq_empty_iff {α : Type u_1} [decidable_eq α] [fintype α] {f : equiv.perm α} :
f.cycle_factors_finset = ∅ ↔ f = 1
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@[simp]
theorem equiv.perm.cycle_factors_finset_one {α : Type u_1} [decidable_eq α] [fintype α] :
1.cycle_factors_finset = ∅
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@[simp]
theorem equiv.perm.cycle_factors_finset_eq_singleton_self_iff {α : Type u_1} [decidable_eq α] [fintype α] {f : equiv.perm α} :
f.cycle_factors_finset = {f} ↔ f.is_cycle
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theorem equiv.perm.is_cycle.cycle_factors_finset_eq_singleton {α : Type u_1} [decidable_eq α] [fintype α] {f : equiv.perm α} (hf : f.is_cycle) :
f.cycle_factors_finset = {f}
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theorem equiv.perm.cycle_factors_finset_eq_singleton_iff {α : Type u_1} [decidable_eq α] [fintype α] {f g : equiv.perm α} :
f.cycle_factors_finset = {g} ↔ f.is_cycle ∧ f = g
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theorem equiv.perm.cycle_factors_finset_injective {α : Type u_1} [decidable_eq α] [fintype α] :
function.injective equiv.perm.cycle_factors_finset

Two permutations f g : perm α have the same cycle factors iff they are the same.

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theorem equiv.perm.disjoint.disjoint_cycle_factors_finset {α : Type u_1} [decidable_eq α] [fintype α] {f g : equiv.perm α} (h : f.disjoint g) :
disjoint f.cycle_factors_finset g.cycle_factors_finset
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theorem equiv.perm.disjoint.cycle_factors_finset_mul_eq_union {α : Type u_1} [decidable_eq α] [fintype α] {f g : equiv.perm α} (h : f.disjoint g) :
(f * g).cycle_factors_finset = f.cycle_factors_finset ∪ g.cycle_factors_finset
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theorem equiv.perm.disjoint_mul_inv_of_mem_cycle_factors_finset {α : Type u_1} [decidable_eq α] [fintype α] {f g : equiv.perm α} (h : f ∈ g.cycle_factors_finset) :
(g * f⁻¹).disjoint f
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theorem equiv.perm.cycle_is_cycle_of {α : Type u_1} [decidable_eq α] [fintype α] {f c : equiv.perm α} {a : α} (ha : a ∈ c.support) (hc : c ∈ f.cycle_factors_finset) :
c = f.cycle_of a

If c is a cycle, a ∈ c.support and c is a cycle of f, then c = f.cycle_of a

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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_cycle → P σ) (induction_disjoint : ∀ (σ τ : equiv.perm β), σ.disjoint τ → σ.is_cycle → P σ → P τ → P (σ * τ)) :
P σ
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theorem equiv.perm.cycle_factors_finset_mul_inv_mem_eq_sdiff {α : Type u_1} [decidable_eq α] [fintype α] {f g : equiv.perm α} (h : f ∈ g.cycle_factors_finset) :
(g * f⁻¹).cycle_factors_finset = g.cycle_factors_finset \ {f}
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theorem equiv.perm.same_cycle.nat_of_mem_support {α : Type u_1} [decidable_eq α] [fintype α] (f : equiv.perm α) {x y : α} (h : f.same_cycle x y) (hx : x ∈ f.support) :
∃ (i : ℕ) (hi' : i < (f.cycle_of x).support.card), ⇑(f ^ i) x = y
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theorem equiv.perm.same_cycle.nat {α : Type u_1} [decidable_eq α] [fintype α] (f : equiv.perm α) {x y : α} (h : f.same_cycle x y) :
∃ (i : ℕ) (hi : 0 < i) (hi' : i ≤ (f.cycle_of x).support.card + 1), ⇑(f ^ i) x = y
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theorem equiv.perm.closure_is_cycle {β : Type u_2} [fintype β] :
subgroup.closure {σ : equiv.perm β | σ.is_cycle} = ⊤
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theorem equiv.perm.closure_cycle_adjacent_swap {α : Type u_1} [decidable_eq α] [fintype α] {σ : equiv.perm α} (h1 : σ.is_cycle) (h2 : σ.support = ⊤) (x : α) :
subgroup.closure {σ, equiv.swap x (⇑σ x)} = ⊤
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theorem equiv.perm.closure_cycle_coprime_swap {α : Type u_1} [decidable_eq α] [fintype α] {n : ℕ} {σ : equiv.perm α} (h0 : n.coprime (fintype.card α)) (h1 : σ.is_cycle) (h2 : σ.support = finset.univ) (x : α) :
subgroup.closure {σ, equiv.swap x (⇑(σ ^ n) x)} = ⊤
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theorem equiv.perm.closure_prime_cycle_swap {α : Type u_1} [decidable_eq α] [fintype α] {σ τ : equiv.perm α} (h0 : nat.prime (fintype.card α)) (h1 : σ.is_cycle) (h2 : σ.support = finset.univ) (h3 : τ.is_swap) :
subgroup.closure {σ, τ} = ⊤
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theorem equiv.perm.is_conj_of_support_equiv {α : Type u_1} [decidable_eq α] [fintype α] {σ τ : equiv.perm α} (f : {x // x ∈ ↑(σ.support)} ≃ {x // x ∈ ↑(τ.support)}) (hf : ∀ (x : α) (hx : x ∈ ↑(σ.support)), ↑(⇑f ⟨⇑σ x, _⟩) = ⇑τ ↑(⇑f ⟨x, hx⟩)) :
is_conj σ τ
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theorem equiv.perm.is_cycle.is_conj {α : Type u_1} [decidable_eq α] [fintype α] {σ τ : equiv.perm α} (hσ : σ.is_cycle) (hτ : τ.is_cycle) (h : σ.support.card = τ.support.card) :
is_conj σ τ
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theorem equiv.perm.is_cycle.is_conj_iff {α : Type u_1} [decidable_eq α] [fintype α] {σ τ : equiv.perm α} (hσ : σ.is_cycle) (hτ : τ.is_cycle) :
is_conj σ τ ↔ σ.support.card = τ.support.card
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@[simp]
theorem equiv.perm.support_conj {α : Type u_1} [decidable_eq α] [fintype α] {σ τ : equiv.perm α} :
((σ * τ) * σ⁻¹).support = finset.map (equiv.to_embedding σ) τ.support
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theorem equiv.perm.card_support_conj {α : Type u_1} [decidable_eq α] [fintype α] {σ τ : equiv.perm α} :
((σ * τ) * σ⁻¹).support.card = τ.support.card
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theorem equiv.perm.disjoint.is_conj_mul {α : Type u_1} [fintype α] {σ τ π ρ : equiv.perm α} (hc1 : is_conj σ π) (hc2 : is_conj τ ρ) (hd1 : σ.disjoint τ) (hd2 : π.disjoint ρ) :
is_conj (σ * τ) (π * ρ)

Fixed points #

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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