instMulDistribMulActionSubtypeMemNormalizer

English

The normalizer N(H) of a subgroup H acts on H by conjugation, giving a natural action by automorphisms.

Русский

Нормализатор N(H) подгруппы H действует на H по сопряжению, образуя естественное действие на пермутации автообразований.

LaTeX

$$N_G(H) acts on H by (g, h) ↦ g h g^{-1}, hence defines a homomorphism N_G(H) → Aut(H).$$

Lean4

/-- The conjugation action of N(H) on H. -/
@[simps]
instance : MulDistribMulAction H.normalizer H
    where
  smul g h := ⟨g * h * g⁻¹, (g.2 h).mp h.2⟩
  one_smul g := by simp [HSMul.hSMul]
  mul_smul := by simp [HSMul.hSMul, mul_assoc]
  smul_one := by simp [HSMul.hSMul]
  smul_mul := by simp [HSMul.hSMul]

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