Fixed point conditions for non-coprime actions

TL;DR

The paper generalizes Glauberman’s fixed-point result to non-coprime actions by examining complements to a normal subgroup $N$ in the semidirect product $G=N\rtimes J$ acting on $\Omega$. Using a cohomological framework with $H^1(J,N)$ and the concept of $J$-invariant elements, it proves that if $N$ is abelian and every Sylow $p$-subgroup of $J$ fixes a point, then a $J$-invariant fixed point exists; this is extended to the case where $N$ is nilpotent and $N\rtimes J$ is supersoluble, via a key lemma that complements of a normal nilpotent subgroup are conjugate whenever they are locally conjugate. The results connect fixed-point existence to the structure and conjugacy of complements, and clarify how cohomological obstructions vanish under these hypotheses, with a note on limitations given by known counterexamples in broader contexts.$

Abstract

In the setting of finite groups, suppose $J$ acts on $N$ via automorphisms so that the induced semidirect product $N\rtimes J$ acts on some non-empty set $Ω$, with $N$ acting transitively. Glauberman proved that if the orders of $J$ and $N$ are coprime, then $J$ fixes a point in $Ω$. We consider the non-coprime case and show that if $N$ is abelian and a Sylow $p$-subgroup of $J$ fixes a point in $Ω$ for each prime $p$, then $J$ fixes a point in $Ω$. We also show that if $N$ is nilpotent, $N\rtimes J$ is supersoluble, and a Sylow $p$-subgroup of $J$ fixes a point in $Ω$ for each prime $p$, then $J$ fixes a point in $Ω$.

Theorems & Definitions (17)

• Lemma 1: Evans and Shin
• Theorem 1
• Corollary 1
• Lemma 2
• Theorem 2
• proof : Proof of Lemma \ref{['lem:ab']}
• proof : Proof of Theorem \ref{['thm:ab']}
• proof : Proof of Corollary \ref{['thm:ab']}
• Proposition 1
• proof