quotientKerEquivRange — Mathlib

English

There is an induced isomorphism between quotients by two equal subgroups M and N: if M = N and they are normal, then G/M ≃* G/N.

Русский

Существует индуцированный изоморфизм между факторами по двумя равными нормальным подгруппам M и N: если M = N и они нормальны, то G/M ≃* G/N.

LaTeX

$$If M = N and M.Normal, then G/M ≃* G/N$$

Lean4

/-- **Noether's first isomorphism theorem** (a definition): the canonical isomorphism between
`G/(ker φ)` to `range φ`. -/
@[to_additive /-- The first isomorphism theorem (a definition): the canonical isomorphism between
    `G/(ker φ)` to `range φ`. -/
    ]
noncomputable def quotientKerEquivRange : G ⧸ ker φ ≃* range φ :=
  MulEquiv.ofBijective (rangeKerLift φ) ⟨rangeKerLift_injective φ, rangeKerLift_surjective φ⟩

Похожие утверждения