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\(\mathscr{P}\)-characters and the structure of finite solvable groups. (English) Zbl 1477.20015

Let \(G\) be a finite group. An irreducible character \(\chi\) of \(G\) is called a \(\mathscr{P}\)-character if it is an irreducible constituent of \(1_H^G\) for some maximal subgroup \(H\) of \(G\); if in addtion \(|G:H|\) is a \(p\)-power for a prime \(p\), then \(\chi\) is called a \(\mathscr{P}_p\)-character. The paper studies some conditions on \(\mathscr{P}\)-characters that are sufficient for the group to be \(p\)-nilpotent or \(p\)-closed. For example, \(G\) is \(p\)-nilpotent if and only if, for every nonlinear \(\mathscr{P}_p\)-character of \(G\) of \(p'\)-degree, \(G/\ker(\chi\)) is \(p\)-nilpotent. Also, if \(G\) is \(p\)-solvable and \(p\) does not divide the codegree of any \(\mathscr{P}\)-character of \(G\), then \(G\) is \(p\)-nilpotent.

MSC:

20C15 Ordinary representations and characters
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D15 Finite nilpotent groups, \(p\)-groups
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References:

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