Group actions related to non-vanishing elements

Abstract

‎We characterize those groups $G$ and vector spaces $V$ such that $V$ is a faithful irreducible $G$-module and such that each $v$ in $V$ is centralized by a $G$-conjugate of a fixed non-identity element of the Fitting subgroup $F(G)$ of $G$‎. ‎We also determine those $V$ and $G$ for which $V$ is a faithful quasi-primitive $G$-module and $F(G)$ has no regular orbit‎. ‎We do use these to show in some cases that a non-vanishing element lies in $F(G)$‎.