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A character theoretic condition for \(F(G)>1\). (English) Zbl 1098.20005

The author answers a question of Y. Berkovich in the affirmative: If \(G\) is a nontrivial finite group with the property that \(\chi(1)^2\) divides \(|G|\) for all irreducible (complex) characters \(\chi\), then \(G\) has a nontrivial normal Abelian subgroup. – The proof uses the classification of finite simple groups and theorems on the existence of \(p\)-blocks of defect zero in these groups.

MSC:

20C15 Ordinary representations and characters
20D25 Special subgroups (Frattini, Fitting, etc.)
20D60 Arithmetic and combinatorial problems involving abstract finite groups
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References:

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