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Finite groups whose irreducible characters vanish only on \(p\)-elements. (English) Zbl 1162.20004

Let \(G\) be a finite group and \(p\) a prime. One of the main results of this paper is the following: Suppose that \(G\) is nonabelian and that whenever \(\chi(g)=0\) for an irreducible character \(\chi\) of \(G\) and some \(g\in G\), then necessarily \(g\) is a \(p\)-element. Then either \(G\) is a \(p\)-group, or \(G/Z(G)\) is a Frobenius group with Frobenius complement of \(p\)-power order and \(Z(G)=O_p(G)\). The result depends of CFSG. A similar result (which, however, does not need CFSG) is shown under the stronger hypothesis that “\(p\)-element” above is replaced by “element of order \(p\)”, and if \(p=2\) in this case, the groups are completely characterized. Some interesting examples are also discussed.

MSC:

20C15 Ordinary representations and characters
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
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