Conjugacy conditions for supersoluble complements of an abelian base and a fixed point result for non-coprime actions

• Published in: Proceedings of the Edinburgh Mathematical Society 2022-11, Vol.65 (4), p.1075-1079
• Main Author: Burkhart, Michael C.
• Format: Article
• Language: English
• Subjects: Conjugates; Hypotheses; Subgroups
• ISSN: 0013-0915; 1464-3839
• DOI: 10.1017/S0013091522000499

We demonstrate that two supersoluble complements of an abelian base in a finite split extension are conjugate if and only if, for each prime $p$ , a Sylow $p$ -subgroup of one complement is conjugate to a Sylow $p$ -subgroup of the other. As a corollary, we find that any two supersoluble complements of an abelian subgroup $N$ in a finite split extension $G$ are conjugate if and only if, for each prime $p$ , there exists a Sylow $p$ -subgroup $S$ of $G$ such that any two complements of $S\cap N$ in $S$ are conjugate in $G$ . In particular, restricting to supersoluble groups allows us to ease D. G. Higman's stipulation that the complements of $S\cap N$ in $S$ be conjugate within $S$ . We then consider group actions and obtain a fixed point result for non-coprime actions analogous to Glauberman's lemma.