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Finite solvable groups with an irreducible character vanishing on just one class of elements. (English) Zbl 1127.20008
Let \(G\) be a finite group. It is well-known that if \(\chi\) is a nonlinear irreducible complex character of \(G\), then \(\chi\) vanishes on some conjugacy class of \(G\). For solvable \(G\), the author is interested in groups for which there exists a \(\chi\) which vanishes on exactly one conjugacy class of \(G\). The main result gives a detailed description of such groups (which is too technical to be reproduced in this review). He also classifies odd order groups which have an irreducible character vanishing on two classes.

MSC:
20C15 Ordinary representations and characters
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
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References:
[1] DOI: 10.1006/jabr.1996.0146 · Zbl 0860.20017
[2] Isaacs I. M., Character Theory of Finite Groups (1976) · Zbl 0337.20005
[3] DOI: 10.1081/AGB-200039286 · Zbl 1094.20013
[4] Zhmud E. M., Soviet Math. Dokl. 20 pp 795– (1979)
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