# On the converse of Gasch\"utz' complement theorem

## TL;DR

This paper extends Gasch"utz' complement theorem by showing that if all Sylow subgroups of a normal subgroup are abelian, then the subgroup has a complement in the larger group, and explores related conditions and counterexamples.

## Contribution

It proves a new sufficient condition for the existence of complements in finite groups, generalizing Gasch"utz' theorem, and analyzes specific cases like metabelian and perfect groups.

## Key findings

- If all Sylow subgroups of N are abelian, N has a complement in G.
- Counterexamples exist when Z(N)∩N'≠1.
- For metabelian N with Z(N)∩N'=1, N has a complement.

## 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 Gasch\"utz 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 Gasch\"utz' theorem holds for N if and only if Inn(N) has a complement in Aut(N).