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 ...
