Theorems · Theorem · group theory

Subgroup.goursat

∀ {G : Type u_1} {H : Type u_2} [inst : Group G] [inst_1 : Group H] {I : Subgroup (G × H)},
  ∃ G' H' M N,
    ∃ (x : M.Normal) (x_1 : N.Normal),
      ∃ e,
        I =
          Subgroup.map (G'.subtype.prodMap H'.subtype)
            (Subgroup.comap ((QuotientGroup.mk' M).prodMap (QuotientGroup.mk' N)) e.toMonoidHom.graph)

Goursat's lemma for an arbitrary subgroup. If I is a subgroup of G × H, then there exist subgroups G' ≤ G, H' ≤ H and normal subgroups M ⊴ G' and N ⊴ H' such that M × N ≤ I and the image of I in G' ⧸ M × H' ⧸ N is the graph of an isomorphism G' ⧸ M ≃ H' ⧸ N.

Defined in

Mathlib.GroupTheory.Goursat

Cited by

0 results in Mathlib

Foundations

Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

GroupGroup