Inequalities concerning the characters of a finite group

Fields, K. L.

doi:10.1090/S0002-9939-1975-0367045-7

Inequalities concerning the characters of a finite group
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Proc. Amer. Math. Soc. 49 (1975), 289-293 Request permission

Erratum: Proc. Amer. Math. Soc. 61 (1976), 186.

Abstract:

Given a finite group we provide explicit bounds (in terms of the group order and numbers of conjugacy classes and involutions) for (a) the number of real valued characters of type ${\mathbf {R}}$; (b) the sum of the degrees of the irreducible characters; (c) the sum of the entries of the character table; (d) the sums (b), (c) restricted to real valued characters. We also provide a bound on the number of elements of order $2n$ in terms of the number of elements of order $n$.

References

Richard Brauer, Representations of finite groups, Lectures on Modern Mathematics, Vol. I, Wiley, New York, 1963, pp. 133–175. MR 0178056
Richard Brauer and K. A. Fowler, On groups of even order, Ann. of Math. (2) 62 (1955), 565–583. MR 74414, DOI 10.2307/1970080
K. L. Fields, On the conjugating representation of a finite group, Proc. Amer. Math. Soc. 34 (1972), 35–37. MR 297889, DOI 10.1090/S0002-9939-1972-0297889-9

Über die Reellen Darstellungen der endlichen Gruppen

Louis Solomon, On the sum of the elements in the character table of a finite group, Proc. Amer. Math. Soc. 12 (1961), 962–963. MR 132783, DOI 10.1090/S0002-9939-1961-0132783-8