On the converse of Gaschütz' complement theorem

Abstract

Let N be a normal subgroup of a finite group G. Let N\le H\le G such that N has a complement in H and (|N|,|G:H|)=1. If N is abelian, a theorem of Gaschütz asserts that N has a complement in G as well. Brandis has asked whether the commutativity of N can be replaced by some weaker property. We prove that N has a complement in G whenever all Sylow subgroups of N are abelian. On the other hand, we construct counterexamples if Z(N)\cap N'\ne 1. For metabelian groups N, the condition Z(N)\cap N'=1 implies the existence of complements. Finally, if N is perfect and centerless, then Gaschütz' theorem holds for N if and only if Inn(N) has a complement in Aut(N).

13 pages, proofs of Shemetkov's and Yonaha's theorems simplified, this version differs significantly from the published paper in J. Group Theory