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

TL;DR

This paper characterizes when supersoluble complements in a finite split extension are conjugate, based on Sylow subgroup conjugacy, and establishes a fixed point theorem for non-coprime group actions.

Contribution

It provides new conjugacy criteria for supersoluble complements and a fixed point result for non-coprime group actions, extending existing theorems.

Findings

Supersoluble complements are conjugate iff Sylow p-subgroups are conjugate for each prime p.

Any two supersoluble complements of an abelian subgroup are conjugate under certain Sylow subgroup conditions.

Established a fixed point theorem for non-coprime group actions analogous to Glauberman's lemma.

Abstract

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.