Abstract
Letd>1 be a proper divisor of the order of a finite groupG and let σ d (G) be the sum of squares of degrees of those irreducible characters whose degrees are not divisible byd. It is easy to see thatd divides σ d (G). The groupsG such that σ d (G) =d coincide with Frobenius groups whose kernel has indexd (see G. Karpilovsky,Group Representations, Volume 1, Part B, North-Holland, Amsterdam, 1992, Theorem 37.5.5). In this note we study the case σ d (G) = 2d in some detail. In particular, ifG is a 2-group, it is of maximal class (Remark 3(b)).
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[B1] Y. Berkovich,Degrees of irreducible characters and normal p-complements, Proceedings of the American Mathematical Society106 (1989), 33–35.
[B2] Y. Berkovich,Finite groups with small sums of degrees of some nonlinear irreducible characters, Journal of Algebra171 (1995), 426–443.
[BZ] Y. Berkovich and E. Zhmud,Characters of Finite Groups, Part 1, American Mathematical Society Translations of Mathematical Monographs 172, Providence, 1998.
[H] B. Huppert,Endliche Gruppen, Bd. 1, Springer, Berlin, 1967.
[I] I. M. Isaacs,Character Theory of Finite Groups, Academic Press, New York, 1976.
[K] G. Karpilovsky,Group Representations, Volume 1, Part B, North-Holland, Amsterdam, 1992.
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The author was supported in part by the Ministry of Absorption of Israel.
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Berkovich, Y. Groups with few characters of small degrees. Isr. J. Math. 110, 325–332 (1999). https://doi.org/10.1007/BF02808187
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DOI: https://doi.org/10.1007/BF02808187