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 $F(G)$.
B. Huppert (1957). Zweifach transitive auflosbare Permutationsgruppen. Math. Z.. 68, 126-150 B. Huppert (1998). Character Theory of Finite Groups. de Gruyter Expositions in Mathematics, Walter de Gruyter and Co., Berlin. 25 B. Huppert and N. Blackburn (1982). Finite Groups II. Springer, Berlin-New York. 44 I. M. Isaacs, G. Navarro and T. Wolf (1999). Finite Groups Elements where no Irreducible Character Vanishes. J. Algebra. 222, 413-423 O. Manz and T. Wolf (1993). Representations of Solvable Groups. London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge. 185 A. Moreto and T. Wolf (2004). Orbit sizes, character degrees, and Sylow subgroups. Adv. Math.. 184, 18-36
Wolf, T. (2014). Group actions related to non-vanishing elements. International Journal of Group Theory, 3(2), 41-51. doi: 10.22108/ijgt.2014.3669
MLA
Thomas Wolf. "Group actions related to non-vanishing elements". International Journal of Group Theory, 3, 2, 2014, 41-51. doi: 10.22108/ijgt.2014.3669
HARVARD
Wolf, T. (2014). 'Group actions related to non-vanishing elements', International Journal of Group Theory, 3(2), pp. 41-51. doi: 10.22108/ijgt.2014.3669
VANCOUVER
Wolf, T. Group actions related to non-vanishing elements. International Journal of Group Theory, 2014; 3(2): 41-51. doi: 10.22108/ijgt.2014.3669
)
