English
If a group G satisfies NormalizerCondition, then every Sylow p-subgroup is normal in G.
Русский
Если у группы G выполняется условие NormalizerCondition, то любая подгруппа Sylow p(G) нормальна в G.
LaTeX
$$$NormalizerCondition(G)\Rightarrow \forall p (P:Sylow_p(G)), P.Normal$$$
Lean4
/-- Torsion groups are closed under extensions. -/
@[to_additive AddIsTorsion.extension_closed /-- Additive torsion groups are closed under extensions. -/
]
theorem extension_closed {f : G →* H} (hN : N = f.ker) (tH : IsTorsion H) (tN : IsTorsion N) : IsTorsion G := fun g =>
by
obtain ⟨ngn, ngnpos, hngn⟩ := (tH <| f g).exists_pow_eq_one
have hmem := MonoidHom.mem_ker.mpr ((f.map_pow g ngn).trans hngn)
lift g ^ ngn to N using hN.symm ▸ hmem with gn h
obtain ⟨nn, nnpos, hnn⟩ := (tN gn).exists_pow_eq_one
exact
isOfFinOrder_iff_pow_eq_one.mpr <|
⟨ngn * nn, mul_pos ngnpos nnpos, by rw [pow_mul, ← h, ← Subgroup.coe_pow, hnn, Subgroup.coe_one]⟩
