English
There exists a canonical isomorphism between a subgroup H of G and its image H.map f when f: G →* N is injective; the construction is driven by image of elements and preserves multiplication.
Русский
Существует каноническое изоморфизм между подгруппой H и её изображением H.map f, когда f: G →* N инъективен; конструкция сохраняет умножение.
LaTeX
$$$H \\cong_{*} H.map f$ for injective $f$$$
Lean4
/-- A subgroup is isomorphic to its image under an injective function. If you have an isomorphism,
use `MulEquiv.subgroupMap` for better definitional equalities. -/
@[to_additive /-- An additive subgroup is isomorphic to its image under an injective function. If you
have an isomorphism, use `AddEquiv.addSubgroupMap` for better definitional equalities. -/
]
noncomputable def equivMapOfInjective (H : Subgroup G) (f : G →* N) (hf : Function.Injective f) : H ≃* H.map f :=
{ Equiv.Set.image f H hf with map_mul' := fun _ _ => Subtype.ext (f.map_mul _ _) }
