card_quotient_normalizer_modEq_card_quotient

English

See item 125183 for the surjectivity claim of the Sylow map under a surjective homomorphism.

Русский

См. пункт 125183 для сюръективности отображения Sylow при сюръективном гомоморфизме.

LaTeX

$$again: Function.Surjective (Sylow.mapSurjective hf)$$

Lean4

/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
  mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Finite G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
    (hH : Nat.card H = p ^ n) :
    Nat.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡ Nat.card (G ⧸ H) [MOD
      p] :=
  by
  rw [← Nat.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
  exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm

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