Group actions related to non-vanishing elements

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).