## 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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