finite_of_injective — Mathlib

English

If f is injective and Sylow p G is finite, then Sylow p H is finite for H domain of f.

Русский

Если f инъективен и Sylow p G конечно, то Sylow p H конечно для области H.

LaTeX

$$$\\text{Finite} (Sylow_p(G)) \\Rightarrow \\text{Finite} (Sylow_p(H))$ under injective f$$

Lean4

/-- If `f : H →* G` is injective, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
theorem finite_of_injective {H : Type*} [Group H] {f : H →* G} (hf : Function.Injective f) [Finite (Sylow p G)] :
    Finite (Sylow p H) :=
  finite_of_ker_is_pGroup (IsPGroup.ker_isPGroup_of_injective hf)

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