Notes on Non-Vanishing Elements of Finite Solvable Groups

Liguo He

Bulletin of the Malaysian Mathematical Society, Tome 35 (2012) no. 1

Résumé

Let $G$ be a finite solvable group. The element $g \in G$ is said to be a non-vanishing element of $G$ if $\chi(g) \neq 0$ for all $\chi \in {\rm Irr\ }(G)$. It is conjectured that all of non-vanishing elements of $G$ lie in its Fitting subgroup $F(G)$. In this note, we prove that this conjecture is true for nilpotent-by-supersolvable groups. Write $\mathscr{V}(G)$ to denote the subgroup generated by all non-vanishing elements of $G$, and $F_n(G)$ the nth term of the ascending Fitting series. It is proved that $\mathscr{V}(F_n(G)) \leq F_{n-1}(G)$ whenever $G$ is solvable. If this conjecture were not true, then it is proved that the minimal counterexample is a solvable primitive permutation group and the more detailed information is presented. Some other related results are proved.

Détail

Classification : 20C15, 20D10.