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Finite groups with a unique nonlinear nonfaithful irreducible character. (English) Zbl 1249.20009
All finite groups, which have exactly one nonlinear irreducible character, were classified by Y. Berkovich, D. Chillag and M. Herzog [Proc. Am. Math. Soc. 115, No. 4, 955-959 (1992; Zbl 0822.20004)]. Based on this result, finite groups, which have exactly one nonlinear nonfaithful irreducible character, are studied in this paper. In particular, it is shown that the only \(p\)-groups with this property are groups of order \(16\) and of nilpotency class \(3\), and a complete classification of nilpotent groups with this property is given.
Reviewer: Michal Kunc (Brno)

MSC:
20C15 Ordinary representations and characters
20D15 Finite nilpotent groups, \(p\)-groups
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