On the converse of Gaschütz' complement theorem

verfasst von

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).

Organisationseinheit(en)

Institut für Algebra, Zahlentheorie und Diskrete Mathematik

Typ

Artikel

Journal

Journal of group theory

Band

26

Seiten

931-949

Anzahl der Seiten

19

ISSN

1433-5883

Publikationsdatum

01.09.2023

Publikationsstatus

Veröffentlicht

Peer-reviewed

Ja

ASJC Scopus Sachgebiete

Algebra und Zahlentheorie

Elektronische Version(en)

https://doi.org/10.48550/arXiv.2303.00254 (Zugang: Offen)
https://doi.org/10.1515/jgth-2022-0178 (Zugang: Geschlossen)