English
For a fixed Sylow p-subgroup P of G, under mild normalizer-centralizer alignments, the conjugation action shows that g0^{-1} g^k g0 ∈ P implies g0^{-1} g^k g0 = g^k, and thus g^index ∈ P.
Русский
Для фиксированной подгруппы П Силова в G, при условии, что нормализатор и центральizer согласованы, сопряжение показывает, что если g0^{-1} g^k g0 ∈ P, то равно g^k, и значит g^{index} ∈ P.
LaTeX
$$$g^{|P|} \in P$ under the stated hypothesis; more generally, conjugation constraints force powers of g to lie in P.$$
Lean4
/-- Given `ϕ : H →* A` from `H : Subgroup G` to a commutative group `A`,
the transfer homomorphism is `transfer ϕ : G →* A`. -/
@[to_additive /-- Given `ϕ : H →+ A` from `H : AddSubgroup G` to an additive commutative group `A`,
the transfer homomorphism is `transfer ϕ : G →+ A`. -/
]
noncomputable def transfer [FiniteIndex H] : G →* A :=
let T : H.LeftTransversal := default
{ toFun := fun g => diff ϕ T (g • T)
map_one' := by rw [one_smul, diff_self]
map_mul' := fun g h => by rw [mul_smul, ← diff_mul_diff, smul_diff_smul] }
