Regular and p-regular orbits of solvable linear groups

Abstract

Let V be a faithful G-module for a finite group G and let p be a prime dividing |G|. An orbit vG for the action of G on V is p-regular if |vG|p=|G:CG(v)|p=|G|p. Zhang asks the following question in Zhang (1993) [8]. Assume that a finite solvable group G acts faithfully and irreducibly on a vector space V over a finite field F. 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 Lü and Cao (2000) [4], Lü 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 Zhangʼs 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.