Regular orbits and p-regular orbits of solvable linear groups

TL;DR

This paper investigates conditions under which solvable groups acting on vector spaces have regular orbits, showing that having p-regular orbits for all primes does not always guarantee a regular orbit, but it does in the case of odd order groups over odd characteristic fields.

Contribution

The paper provides counterexamples to Zhang's question and proves that for odd order solvable groups over odd characteristic fields, p-regular orbits imply the existence of a regular orbit.

Findings

Counterexamples show the general answer is no.

For odd order groups over odd characteristic fields, p-regular orbits imply a regular orbit.

Abstract

Let $V$ be a faithful $G$-module for a finite group $G$ and let $p$ be a prime dividing $|G|$. An orbit $v^G$ for the action of $G$ on $V$ is $p$-regular if $|v^G|_p=|G:\bC_G(v)|_p=|G|_p$. Zhang asks the following question in \cite{Zhang}. Assume that a finite solvable group $G$ acts faithfully and irreducibly on a vector space $V$ over a finite field $\FF$. If $G$ has a $p$-regular orbit for every prime $p$ dividing $|G|$, is it true that $G$ will have a regular orbit on $V$? In \cite{LuCao}, L\"{u} and Cao construct an example showing that the answer to this question is no, however the example itself is not correct. In this paper, we study Zhang's question in detail. We construct examples showing that the answer to this question is no in general. We also prove the following result. Assume a finite solvable group $G$ of odd order acts faithfully and irreducibly on a vector space $V$ over a field of odd characteristic. If $G$ has a $p$-regular orbit for every prime $p$ dividing $|G|$, then $G$ will have a regular orbit on $V$.