Mathematics – Group Theory
Scientific paper
2011-01-18
Mathematics
Group Theory
Scientific paper
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 $V$.
Keller Thomas Michael
Yang Yong
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