Year: 2026
Type: preprint
Abstract: A classical theorem of Gaschütz states that an abelian normal subgroup N of a finite group G has a complement in G whenever it has a complement in some subgroup H with $N \leq H \leq G$ and gcd(|N|, [G:H]) = 1. We characterize the finite groups N for which this theorem remains valid without the abelian hypothesis: this is the case if and only if $Z(N) \cap N' = 1$ and N has a complement in a quotient of the holomorph Hol(N) by a diagonal copy of N'. This completes a previous paper by the second author.
Source: arXiv (Cornell University)
Authors: Aivazidis, Stefanos, Sambale, Benjamin
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204843355