isCyclic_of_coprime_card_ker — Mathlib

English

The index of the range of powMonoidHom d equals gcd(|G|, d).

Русский

Индекс образа powMonoidHom d равен gcd(|G|, d).

LaTeX

$$(powMonoidHom d).range.index = Nat.card G gcd d$$

Lean4

@[to_additive]
theorem isCyclic_of_coprime_card_ker {M N : Type*} [CommGroup M] [Group N] (f : M →* N)
    (h : (Nat.card f.ker).Coprime (Nat.card N)) [IsCyclic f.ker] [hN : IsCyclic N] (hf : Function.Surjective f) :
    IsCyclic M := by
  rw [← Subgroup.topEquiv.isCyclic, ← f.range_eq_top.mpr hf] at hN
  rw [← Subgroup.card_top (G := N), ← f.range_eq_top.mpr hf] at h
  exact isCyclic_of_coprime_card_range_card_ker f h

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