On the converse of Gaschütz' complement theorem

authored by

Benjamin Sambale

Abstract

Let N be a normal subgroup of a finite group G. Let N ≤ H ≤ 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) ∩ N ′ ≠ 1. For metabelian groups N, the condition Z (N) ≠ 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).

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

Type

Article

Journal

Journal of group theory

Volume

26

Pages

931-949

No. of pages

19

ISSN

1433-5883

Publication date

01.09.2023

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

Algebra and Number Theory

Electronic version(s)

https://doi.org/10.48550/arXiv.2303.00254 (Access: Open)
https://doi.org/10.1515/jgth-2022-0178 (Access: Closed)