The object of this note is to give simpler proofs of a splitting theorem of Gaschütz [1] and a related theorem for groups with operators by using cross-homomorphisms (1-cocycles) instead of 2-cohomology.

We recall that a cross-homomorphism or 1-cocycle from a group E to an abelian normal subgroup N of E is a map f from E to N such that f(ab)=(f(a))b f(b) for all a, b ∈ E where superscript denotes conjugation. Cocycles f and h are equivalent if for some n ∈ N have h(e)=n e f(e)n -1 for all e ∈ E.