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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A class of finite groups admitting certain sharp characters. II

Authors: Takashi Matsuhisa and Hiroyoshi Yamaki
Journal: Proc. Amer. Math. Soc. 110 (1990), 1-5
MSC: Primary 20C15; Secondary 20B20
MathSciNet review: 1030736
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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

$\displaystyle \frac{{a(l)}} {{\vert G\vert}}\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^\char93 } = \operatorname{Im} \chi \backslash \left\{ {\chi (1)} \right\}$ as follows: (i) $ {L^\char93 }$ consists of roots of unity together with $ 0 \in {\mathbf{Z}}$, and (ii) $ {L^\char93 } = \left\{ {0,{l_1},{l_2}, \ldots ,{l_t}} \right\}$ with $ (\vert G\vert,{l_i}) = 1({l_i} \in {\mathbf{Z}},t \geq 2)$. In both cases, if $ (G,\chi ,0)$ is sharp of type $ {L^\char93 }$ as above, $ G$ is either a sharply $ 3$-transitive permutation group or a $ 2$-transitive Forbenius group.

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Keywords: Characters of finite groups, Frobenius groups, transitive permutation groups
Article copyright: © Copyright 1990 American Mathematical Society