A class of finite groups admitting certain sharp characters. II

Abstract:

A triple $(G,\chi ,l)$ of a finite group $G$ with a faithful character $\chi$ and a character value $l$ is called sharp if \[ \frac {{a(l)}} {{|G|}}\prod \limits _{k \in L\backslash \{ l\} } {(l - k)} \] is a unit in algebraic integers (where $l \in L$, the set of character values of $\chi$ , and $a(l)$ is the number of $x \in G$ with $\chi (x) = l$, which generalizes the notion of sharply multiply transitive permutation groups. In this note, we shall determine sharp triples admitting the character values ${L^\# } = \operatorname {Im} \chi \backslash \left \{ {\chi (1)} \right \}$ as follows: (i) ${L^\# }$ consists of roots of unity together with $0 \in {\mathbf {Z}}$, and (ii) ${L^\# } = \left \{ {0,{l_1},{l_2}, \ldots ,{l_t}} \right \}$ with $(|G|,{l_i}) = 1({l_i} \in {\mathbf {Z}},t \geq 2)$. In both cases, if $(G,\chi ,0)$ is sharp of type ${L^\# }$ as above, $G$ is either a sharply $3$-transitive permutation group or a $2$-transitive Forbenius group.