Fixed point conditions for non-coprime actions

TL;DR

This paper extends fixed point theorems for group actions from the coprime case to certain non-coprime scenarios, focusing on abelian and nilpotent groups.

Contribution

It generalizes Glauberman's fixed point result to non-coprime actions when the acting group has specific structural properties.

Findings

If N is abelian and Sylow p-subgroups of J fix a point, then J fixes a point.

For nilpotent N and supersoluble N⋉J, similar fixed point results hold.

The results apply to broader classes of group actions beyond coprime conditions.

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 $\Omega$, with $N$ acting transitively. Glauberman proved that if the orders of $J$ and $N$ are coprime, then $J$ fixes a point in $\Omega$. We consider the non-coprime case and show that if $N$ is abelian and a Sylow $p$-subgroup of $J$ fixes a point in $\Omega$ for each prime $p$, then $J$ fixes a point in $\Omega$. We also show that if $N$ is nilpotent, $N\rtimes J$ is supersoluble, and a Sylow $p$-subgroup of $J$ fixes a point in $\Omega$ for each prime $p$, then $J$ fixes a point in $\Omega$.