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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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