Cycle factors of a permutation

Let β be a Fintype and f : Equiv.Perm β.

• Equiv.Perm.cycleOf: f.cycleOf x is the cycle of f that x belongs to.
• Equiv.Perm.cycleFactors: f.cycleFactors is a list of disjoint cyclic permutations that multiply to f.

cycleOf

def Equiv.Perm.cycleOf {α : Type u_2} (f : Equiv.Perm α) [DecidableRel f.SameCycle] (x : α) : Equiv.Perm α

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

Equations

• f.cycleOf x = Equiv.Perm.ofSubtype (f.subtypePerm ⋯)

theorem Equiv.Perm.cycleOf_apply {α : Type u_2} (f : Equiv.Perm α) [DecidableRel f.SameCycle] (x y : α) : (f.cycleOf x) y = if f.SameCycle x y then f y else y

theorem Equiv.Perm.cycleOf_inv {α : Type u_2} (f : Equiv.Perm α) [DecidableRel f.SameCycle] (x : α) : (f.cycleOf x)⁻¹ = f⁻¹.cycleOf x

@[simp]

theorem Equiv.Perm.cycleOf_pow_apply_self {α : Type u_2} (f : Equiv.Perm α) [DecidableRel f.SameCycle] (x : α) (n : ℕ) : (f.cycleOf x ^ n) x = (f ^ n) x

@[simp]

theorem Equiv.Perm.cycleOf_zpow_apply_self {α : Type u_2} (f : Equiv.Perm α) [DecidableRel f.SameCycle] (x : α) (n : ℤ) : (f.cycleOf x ^ n) x = (f ^ n) x

theorem Equiv.Perm.SameCycle.cycleOf_apply {α : Type u_2} {f : Equiv.Perm α} {x y : α} [DecidableRel f.SameCycle] : f.SameCycle x y → (f.cycleOf x) y = f y

theorem Equiv.Perm.cycleOf_apply_of_not_sameCycle {α : Type u_2} {f : Equiv.Perm α} {x y : α} [DecidableRel f.SameCycle] : ¬f.SameCycle x y → (f.cycleOf x) y = y

theorem Equiv.Perm.SameCycle.cycleOf_eq {α : Type u_2} {f : Equiv.Perm α} {x y : α} [DecidableRel f.SameCycle] (h : f.SameCycle x y) : f.cycleOf x = f.cycleOf y

@[simp]

theorem Equiv.Perm.cycleOf_apply_apply_zpow_self {α : Type u_2} (f : Equiv.Perm α) [DecidableRel f.SameCycle] (x : α) (k : ℤ) : (f.cycleOf x) ((f ^ k) x) = (f ^ (k + 1)) x

@[simp]

theorem Equiv.Perm.cycleOf_apply_apply_pow_self {α : Type u_2} (f : Equiv.Perm α) [DecidableRel f.SameCycle] (x : α) (k : ℕ) : (f.cycleOf x) ((f ^ k) x) = (f ^ (k + 1)) x

@[simp]

theorem Equiv.Perm.cycleOf_apply_apply_self {α : Type u_2} (f : Equiv.Perm α) [DecidableRel f.SameCycle] (x : α) : (f.cycleOf x) (f x) = f (f x)

@[simp]

theorem Equiv.Perm.cycleOf_apply_self {α : Type u_2} (f : Equiv.Perm α) [DecidableRel f.SameCycle] (x : α) : (f.cycleOf x) x = f x

theorem Equiv.Perm.IsCycle.cycleOf_eq {α : Type u_2} {f : Equiv.Perm α} {x : α} [DecidableRel f.SameCycle] (hf : f.IsCycle) (hx : f x ≠ x) : f.cycleOf x = f

@[simp]

theorem Equiv.Perm.cycleOf_eq_one_iff {α : Type u_2} {x : α} (f : Equiv.Perm α) [DecidableRel f.SameCycle] : f.cycleOf x = 1 ↔ f x = x

@[simp]

theorem Equiv.Perm.cycleOf_self_apply {α : Type u_2} (f : Equiv.Perm α) [DecidableRel f.SameCycle] (x : α) : f.cycleOf (f x) = f.cycleOf x

@[simp]

theorem Equiv.Perm.cycleOf_self_apply_pow {α : Type u_2} (f : Equiv.Perm α) [DecidableRel f.SameCycle] (n : ℕ) (x : α) : f.cycleOf ((f ^ n) x) = f.cycleOf x

@[simp]

theorem Equiv.Perm.cycleOf_self_apply_zpow {α : Type u_2} (f : Equiv.Perm α) [DecidableRel f.SameCycle] (n : ℤ) (x : α) : f.cycleOf ((f ^ n) x) = f.cycleOf x

theorem Equiv.Perm.IsCycle.cycleOf {α : Type u_2} {f : Equiv.Perm α} {x : α} [DecidableRel f.SameCycle] [DecidableEq α] (hf : f.IsCycle) : f.cycleOf x = if f x = x then 1 else f

theorem Equiv.Perm.cycleOf_one {α : Type u_2} [DecidableRel (Equiv.Perm.SameCycle 1)] (x : α) : Equiv.Perm.cycleOf 1 x = 1

theorem Equiv.Perm.isCycle_cycleOf {α : Type u_2} {x : α} (f : Equiv.Perm α) [DecidableRel f.SameCycle] (hx : f x ≠ x) : (f.cycleOf x).IsCycle

@[simp]

theorem Equiv.Perm.two_le_card_support_cycleOf_iff {α : Type u_2} {f : Equiv.Perm α} {x : α} [DecidableEq α] [Fintype α] : 2 ≤ (f.cycleOf x).support.card ↔ f x ≠ x

@[simp]

theorem Equiv.Perm.support_cycleOf_nonempty {α : Type u_2} {f : Equiv.Perm α} {x : α} [DecidableEq α] [Fintype α] : (f.cycleOf x).support.Nonempty ↔ f x ≠ x

@[deprecated Equiv.Perm.support_cycleOf_nonempty]

theorem Equiv.Perm.card_support_cycleOf_pos_iff {α : Type u_2} {f : Equiv.Perm α} {x : α} [DecidableEq α] [Fintype α] : 0 < (f.cycleOf x).support.card ↔ f x ≠ x

theorem Equiv.Perm.pow_mod_orderOf_cycleOf_apply {α : Type u_2} (f : Equiv.Perm α) [DecidableRel f.SameCycle] (n : ℕ) (x : α) : (f ^ (n % orderOf (f.cycleOf x))) x = (f ^ n) x

theorem Equiv.Perm.cycleOf_mul_of_apply_right_eq_self {α : Type u_2} {f g : Equiv.Perm α} [DecidableRel f.SameCycle] [DecidableRel (f * g).SameCycle] (h : Commute f g) (x : α) (hx : g x = x) : (f * g).cycleOf x = f.cycleOf x

theorem Equiv.Perm.Disjoint.cycleOf_mul_distrib {α : Type u_2} {f g : Equiv.Perm α} [DecidableRel f.SameCycle] [DecidableRel g.SameCycle] [DecidableRel (f * g).SameCycle] [DecidableRel (g * f).SameCycle] (h : f.Disjoint g) (x : α) : (f * g).cycleOf x = f.cycleOf x * g.cycleOf x

theorem Equiv.Perm.support_cycleOf_eq_nil_iff {α : Type u_2} {f : Equiv.Perm α} {x : α} [DecidableEq α] [Fintype α] : (f.cycleOf x).support = ∅ ↔ x ∉ f.support

theorem Equiv.Perm.support_cycleOf_le {α : Type u_2} [DecidableEq α] [Fintype α] (f : Equiv.Perm α) (x : α) : (f.cycleOf x).support ≤ f.support

theorem Equiv.Perm.mem_support_cycleOf_iff {α : Type u_2} {f : Equiv.Perm α} {x y : α} [DecidableEq α] [Fintype α] : y ∈ (f.cycleOf x).support ↔ f.SameCycle x y ∧ x ∈ f.support

theorem Equiv.Perm.mem_support_cycleOf_iff' {α : Type u_2} {f : Equiv.Perm α} {x y : α} (hx : f x ≠ x) [DecidableEq α] [Fintype α] : y ∈ (f.cycleOf x).support ↔ f.SameCycle x y

theorem Equiv.Perm.SameCycle.mem_support_iff {α : Type u_2} {x y : α} {f : Equiv.Perm α} [DecidableEq α] [Fintype α] (h : f.SameCycle x y) : x ∈ f.support ↔ y ∈ f.support

theorem Equiv.Perm.pow_mod_card_support_cycleOf_self_apply {α : Type u_2} [DecidableEq α] [Fintype α] (f : Equiv.Perm α) (n : ℕ) (x : α) : (f ^ (n % (f.cycleOf x).support.card)) x = (f ^ n) x

theorem Equiv.Perm.isCycle_cycleOf_iff {α : Type u_2} {x : α} (f : Equiv.Perm α) [DecidableRel f.SameCycle] : (f.cycleOf x).IsCycle ↔ f x ≠ x

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

theorem Equiv.Perm.isCycleOn_support_cycleOf {α : Type u_2} [DecidableEq α] [Fintype α] (f : Equiv.Perm α) (x : α) : f.IsCycleOn ↑(f.cycleOf x).support

theorem Equiv.Perm.SameCycle.exists_pow_eq_of_mem_support {α : Type u_2} {x y : α} {f : Equiv.Perm α} [DecidableEq α] [Fintype α] (h : f.SameCycle x y) (hx : x ∈ f.support) : ∃ i < (f.cycleOf x).support.card, (f ^ i) x = y

theorem Equiv.Perm.SameCycle.exists_pow_eq {α : Type u_2} {x y : α} [DecidableEq α] [Fintype α] (f : Equiv.Perm α) (h : f.SameCycle x y) : ∃ (i : ℕ), 0 < i ∧ i ≤ (f.cycleOf x).support.card + 1 ∧ (f ^ i) x = y

theorem Equiv.Perm.zpow_eq_zpow_on_iff {α : Type u_2} [DecidableEq α] [Fintype α] (g : Equiv.Perm α) {m n : ℤ} {x : α} (hx : g x ≠ x) : (g ^ m) x = (g ^ n) x ↔ m % ↑(g.cycleOf x).support.card = n % ↑(g.cycleOf x).support.card

cycleFactors

def Equiv.Perm.cycleFactorsAux {α : Type u_2} [DecidableEq α] [Fintype α] (l : List α) (f : Equiv.Perm α) (h : ∀ {x : α}, f x ≠ x → x ∈ l) : { l : List (Equiv.Perm α) // l.prod = f ∧ (∀ g ∈ l, g.IsCycle) ∧ 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

• One or more equations did not get rendered due to their size.
• Equiv.Perm.cycleFactorsAux [] f h_2 = ⟨[], ⋯⟩

theorem Equiv.Perm.mem_list_cycles_iff {α : Type u_4} [Finite α] {l : List (Equiv.Perm α)} (h1 : ∀ σ ∈ l, σ.IsCycle) (h2 : List.Pairwise Equiv.Perm.Disjoint l) {σ : Equiv.Perm α} : σ ∈ l ↔ σ.IsCycle ∧ ∀ (a : α), σ a ≠ a → σ a = l.prod a

theorem Equiv.Perm.list_cycles_perm_list_cycles {α : Type u_4} [Finite α] {l₁ l₂ : List (Equiv.Perm α)} (h₀ : l₁.prod = l₂.prod) (h₁l₁ : ∀ σ ∈ l₁, σ.IsCycle) (h₁l₂ : ∀ σ ∈ l₂, σ.IsCycle) (h₂l₁ : List.Pairwise Equiv.Perm.Disjoint l₁) (h₂l₂ : List.Pairwise Equiv.Perm.Disjoint l₂) : l₁.Perm l₂

def Equiv.Perm.cycleFactors {α : Type u_2} [Fintype α] [LinearOrder α] (f : Equiv.Perm α) : { l : List (Equiv.Perm α) // l.prod = f ∧ (∀ g ∈ l, g.IsCycle) ∧ List.Pairwise Equiv.Perm.Disjoint l }

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

Equations

• f.cycleFactors = Equiv.Perm.cycleFactorsAux (Finset.sort (fun (x1 x2 : α) => x1 ≤ x2) Finset.univ) f ⋯

def Equiv.Perm.truncCycleFactors {α : Type u_2} [DecidableEq α] [Fintype α] (f : Equiv.Perm α) : Trunc { l : List (Equiv.Perm α) // l.prod = f ∧ (∀ g ∈ l, g.IsCycle) ∧ 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

• One or more equations did not get rendered due to their size.

def Equiv.Perm.cycleFactorsFinset {α : Type u_2} [DecidableEq 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.

Equations

• One or more equations did not get rendered due to their size.

theorem Equiv.Perm.cycleFactorsFinset_eq_list_toFinset {α : Type u_2} [DecidableEq α] [Fintype α] {σ : Equiv.Perm α} {l : List (Equiv.Perm α)} (hn : l.Nodup) : σ.cycleFactorsFinset = l.toFinset ↔ (∀ f ∈ l, f.IsCycle) ∧ List.Pairwise Equiv.Perm.Disjoint l ∧ l.prod = σ

theorem Equiv.Perm.cycleFactorsFinset_eq_finset {α : Type u_2} [DecidableEq α] [Fintype α] {σ : Equiv.Perm α} {s : Finset (Equiv.Perm α)} : σ.cycleFactorsFinset = s ↔ (∀ f ∈ s, f.IsCycle) ∧ ∃ (h : (↑s).Pairwise Equiv.Perm.Disjoint), s.noncommProd id ⋯ = σ

theorem Equiv.Perm.cycleFactorsFinset_pairwise_disjoint {α : Type u_2} [DecidableEq α] [Fintype α] (f : Equiv.Perm α) : (↑f.cycleFactorsFinset).Pairwise Equiv.Perm.Disjoint

theorem Equiv.Perm.cycleFactorsFinset_mem_commute {α : Type u_2} [DecidableEq α] [Fintype α] (f : Equiv.Perm α) : (↑f.cycleFactorsFinset).Pairwise Commute

theorem Equiv.Perm.cycleFactorsFinset_noncommProd {α : Type u_2} [DecidableEq α] [Fintype α] (f : Equiv.Perm α) (comm : (↑f.cycleFactorsFinset).Pairwise Commute := ⋯) : f.cycleFactorsFinset.noncommProd id comm = f

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

theorem Equiv.Perm.mem_cycleFactorsFinset_iff {α : Type u_2} [DecidableEq α] [Fintype α] {f p : Equiv.Perm α} : p ∈ f.cycleFactorsFinset ↔ p.IsCycle ∧ ∀ a ∈ p.support, p a = f a

theorem Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff {α : Type u_2} [DecidableEq α] [Fintype α] {f : Equiv.Perm α} {x : α} : f.cycleOf x ∈ f.cycleFactorsFinset ↔ x ∈ f.support

theorem Equiv.Perm.cycleOf_ne_one_iff_mem_cycleFactorsFinset {α : Type u_2} [DecidableEq α] [Fintype α] {g : Equiv.Perm α} {x : α} : g.cycleOf x ≠ 1 ↔ g.cycleOf x ∈ g.cycleFactorsFinset

theorem Equiv.Perm.mem_cycleFactorsFinset_support_le {α : Type u_2} [DecidableEq α] [Fintype α] {p f : Equiv.Perm α} (h : p ∈ f.cycleFactorsFinset) : p.support ≤ f.support

theorem Equiv.Perm.support_zpowers_of_mem_cycleFactorsFinset_le {α : Type u_2} [DecidableEq α] [Fintype α] {g : Equiv.Perm α} {c : { x : Equiv.Perm α // x ∈ g.cycleFactorsFinset }} (v : ↥(Subgroup.zpowers ↑c)) : (↑v).support ≤ g.support

theorem Equiv.Perm.mem_support_iff_mem_support_of_mem_cycleFactorsFinset {α : Type u_2} [DecidableEq α] [Fintype α] {g : Equiv.Perm α} {x : α} : x ∈ g.support ↔ ∃ c ∈ g.cycleFactorsFinset, x ∈ c.support

theorem Equiv.Perm.cycleFactorsFinset_eq_empty_iff {α : Type u_2} [DecidableEq α] [Fintype α] {f : Equiv.Perm α} : f.cycleFactorsFinset = ∅ ↔ f = 1

@[simp]

theorem Equiv.Perm.cycleFactorsFinset_one {α : Type u_2} [DecidableEq α] [Fintype α] : Equiv.Perm.cycleFactorsFinset 1 = ∅

@[simp]

theorem Equiv.Perm.cycleFactorsFinset_eq_singleton_self_iff {α : Type u_2} [DecidableEq α] [Fintype α] {f : Equiv.Perm α} : f.cycleFactorsFinset = {f} ↔ f.IsCycle

theorem Equiv.Perm.IsCycle.cycleFactorsFinset_eq_singleton {α : Type u_2} [DecidableEq α] [Fintype α] {f : Equiv.Perm α} (hf : f.IsCycle) : f.cycleFactorsFinset = {f}

theorem Equiv.Perm.cycleFactorsFinset_eq_singleton_iff {α : Type u_2} [DecidableEq α] [Fintype α] {f g : Equiv.Perm α} : f.cycleFactorsFinset = {g} ↔ f.IsCycle ∧ f = g

theorem Equiv.Perm.cycleFactorsFinset_injective {α : Type u_2} [DecidableEq α] [Fintype α] : Function.Injective Equiv.Perm.cycleFactorsFinset

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

theorem Equiv.Perm.Disjoint.disjoint_cycleFactorsFinset {α : Type u_2} [DecidableEq α] [Fintype α] {f g : Equiv.Perm α} (h : f.Disjoint g) : Disjoint f.cycleFactorsFinset g.cycleFactorsFinset

theorem Equiv.Perm.Disjoint.cycleFactorsFinset_mul_eq_union {α : Type u_2} [DecidableEq α] [Fintype α] {f g : Equiv.Perm α} (h : f.Disjoint g) : (f * g).cycleFactorsFinset = f.cycleFactorsFinset ∪ g.cycleFactorsFinset

theorem Equiv.Perm.disjoint_mul_inv_of_mem_cycleFactorsFinset {α : Type u_2} [DecidableEq α] [Fintype α] {f g : Equiv.Perm α} (h : f ∈ g.cycleFactorsFinset) : (g * f⁻¹).Disjoint f

theorem Equiv.Perm.cycle_is_cycleOf {α : Type u_2} [DecidableEq α] [Fintype α] {f c : Equiv.Perm α} {a : α} (ha : a ∈ c.support) (hc : c ∈ f.cycleFactorsFinset) : c = f.cycleOf a

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

theorem Equiv.Perm.isCycleOn_support_of_mem_cycleFactorsFinset {α : Type u_2} [DecidableEq α] [Fintype α] {g c : Equiv.Perm α} (hc : c ∈ g.cycleFactorsFinset) : g.IsCycleOn ↑c.support

theorem Equiv.Perm.eq_cycleOf_of_mem_cycleFactorsFinset_iff {α : Type u_2} [DecidableEq α] [Fintype α] (g c : Equiv.Perm α) (hc : c ∈ g.cycleFactorsFinset) (x : α) : c = g.cycleOf x ↔ x ∈ c.support

theorem Equiv.Perm.mem_cycleFactorsFinset_conj {α : Type u_2} [DecidableEq α] [Fintype α] (g k c : Equiv.Perm α) : k * c * k⁻¹ ∈ (k * g * k⁻¹).cycleFactorsFinset ↔ c ∈ g.cycleFactorsFinset

A permutation c is a cycle of g iff k * c * k⁻¹ is a cycle of k * g * k⁻¹

theorem Equiv.Perm.commute_of_mem_cycleFactorsFinset_commute {α : Type u_2} [DecidableEq α] [Fintype α] (k g : Equiv.Perm α) (hk : ∀ c ∈ g.cycleFactorsFinset, Commute k c) : Commute k g

If a permutation commutes with every cycle of g, then it commutes with g

NB. The converse is false. Commuting with every cycle of g means that we belong to the kernel of the action of Equiv.Perm α on g.cycleFactorsFinset

theorem Equiv.Perm.self_mem_cycle_factors_commute {α : Type u_2} [DecidableEq α] [Fintype α] {g c : Equiv.Perm α} (hc : c ∈ g.cycleFactorsFinset) : Commute c g

The cycles of a permutation commute with it

theorem Equiv.Perm.mem_support_cycle_of_cycle {α : Type u_2} [DecidableEq α] [Fintype α] {g d c : Equiv.Perm α} (hc : c ∈ g.cycleFactorsFinset) (hd : d ∈ g.cycleFactorsFinset) (x : α) : x ∈ c.support ↔ d x ∈ c.support

If c and d are cycles of g, then d stabilizes the support of c

theorem Equiv.Perm.mem_cycleFactorsFinset_support {α : Type u_2} [DecidableEq α] [Fintype α] {g c : Equiv.Perm α} (hc : c ∈ g.cycleFactorsFinset) (a : α) : a ∈ c.support ↔ g a ∈ c.support

If a permutation is a cycle of g, then its support is invariant under g

theorem Equiv.Perm.cycle_induction_on {β : Type u_3} [Finite β] (P : Equiv.Perm β → Prop) (σ : Equiv.Perm β) (base_one : P 1) (base_cycles : ∀ (σ : Equiv.Perm β), σ.IsCycle → P σ) (induction_disjoint : ∀ (σ τ : Equiv.Perm β), σ.Disjoint τ → σ.IsCycle → P σ → P τ → P (σ * τ)) : P σ

theorem Equiv.Perm.cycleFactorsFinset_mul_inv_mem_eq_sdiff {α : Type u_2} [DecidableEq α] [Fintype α] {f g : Equiv.Perm α} (h : f ∈ g.cycleFactorsFinset) : (g * f⁻¹).cycleFactorsFinset = g.cycleFactorsFinset \ {f}

theorem Equiv.Perm.IsCycle.forall_commute_iff {α : Type u_2} [DecidableEq α] [Fintype α] (g z : Equiv.Perm α) : (∀ c ∈ g.cycleFactorsFinset, Commute z c) ↔ ∀ c ∈ g.cycleFactorsFinset, ∃ (hc : ∀ (x : α), x ∈ c.support ↔ z x ∈ c.support), Equiv.Perm.ofSubtype (z.subtypePerm hc) ∈ Subgroup.zpowers c

theorem Equiv.Perm.subtypePerm_on_cycleFactorsFinset {α : Type u_2} [DecidableEq α] [Fintype α] {g c : Equiv.Perm α} (hc : c ∈ g.cycleFactorsFinset) : g.subtypePerm ⋯ = c.subtypePermOfSupport

A permutation restricted to the support of a cycle factor is that cycle factor

theorem Equiv.Perm.commute_iff_of_mem_cycleFactorsFinset {α : Type u_2} [DecidableEq α] [Fintype α] {g k c : Equiv.Perm α} (hc : c ∈ g.cycleFactorsFinset) : Commute k c ↔ ∃ (hc' : ∀ (x : α), x ∈ c.support ↔ k x ∈ c.support), k.subtypePerm hc' ∈ Subgroup.zpowers (g.subtypePerm ⋯)