Gagola, Stephen M. jun. A character theoretic condition for \(F(G)>1\). (English) Zbl 1098.20005 Commun. Algebra 33, No. 5, 1369-1382 (2005). 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. Reviewer: Wolfgang Müller (Bayreuth) Cited in 7 Documents MSC: 20C15 Ordinary representations and characters 20D25 Special subgroups (Frattini, Fitting, etc.) 20D60 Arithmetic and combinatorial problems involving abstract finite groups Keywords:Fitting subgroup; irreducible characters; finite groups; normal Abelian subgroups PDF BibTeX XML Cite \textit{S. M. Gagola jun.}, Commun. Algebra 33, No. 5, 1369--1382 (2005; Zbl 1098.20005) Full Text: DOI OpenURL References: [1] Conway J., The Atlas of Finite Groups (1985) [2] Gagola S. M., Communications in Algebra 27 pp 1053– (1999) · Zbl 0929.20010 [3] Granville A., TAMS 348 pp 331– (1996) · Zbl 0855.20007 [4] Isaacs I. M., Character Theory of Finite Groups (1976) · Zbl 0337.20005 [5] James G., The Representation Theory of the Symmetric Group (1981) [6] Michler G., J. Algebra 104 pp 220– (1986) · Zbl 0608.20005 [7] Willems W., J. Algebra 113 pp 511– (1988) · Zbl 0653.20014 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.