equivOfComm — Mathlib

English

For a group isomorphism e: G ≅ H and x ∈ G, the abelianization congruence sends the image of x to the image of e(x); equivalently, [x] maps to [e(x)].

Русский

Для изоморфизма e: G ≅ H и элемента x ∈ G, абелизационная конгруэнция отправляет образ x в образ e(x).

LaTeX

$$$ e^{ab}([x]) = [e(x)], \ x \in G. $$$

Lean4

/-- An Abelian group is equivalent to its own abelianization. -/
@[simps]
def equivOfComm {H : Type*} [CommGroup H] : H ≃* Abelianization H :=
  { Abelianization.of with
    toFun := Abelianization.of
    invFun := Abelianization.lift (MonoidHom.id H)
    right_inv := by
      rintro ⟨a⟩
      rfl }

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