nat_card_centralizer — Mathlib

English

Equality of ranges for alternate formulations of toPermHom'

Русский

Равенство образов для различных формулировок toPermHom'.

LaTeX

$$range_toPermHom_eq_range_toPermHom'$$

Lean4

/-- Cardinality of the centralizer in `Equiv.Perm α` of a permutation given `cycleType` -/
theorem nat_card_centralizer :
    Nat.card (centralizer { g }) =
      (Fintype.card α - g.cycleType.sum)! * g.cycleType.prod * (∏ n ∈ g.cycleType.toFinset, (g.cycleType.count n)!) :=
  by
  rw [← (toPermHom g).ker.card_mul_index, index_ker, nat_card_range_toPermHom, ← kerParam_range_card, ←
    Nat.card_eq_fintype_card, kerParam_range_eq, card_subtype]

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