On a congruence of Blichfeldt concerning the order of finite groups

Author:
David Chillag

Journal:
Proc. Amer. Math. Soc. **136** (2008), 1961-1966

MSC (2000):
Primary 20G15

DOI:
https://doi.org/10.1090/S0002-9939-08-09380-5

Published electronically:
February 14, 2008

MathSciNet review:
2383502

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Abstract | References | Similar Articles | Additional Information

Abstract: We show that if is a finite group, a conjugacy class of and , are the distinct elements in the multiset (here is the value of on any element of ), then

We also observe that in Blichfeldt's congruence can be replaced, with a minor adjustment, by any rational value of . A similar change can be done to the first congruence above.

**1.**Arad, Z.; Stavi, J.; Herzog, M. Powers and products of conjugacy classes in groups. Products of Conjugacy Classes in Groups, Lecture Notes in Math., 1112 (1985) 6-51, Springer, Berlin. MR**783068****2.**Blichfeldt, H. F., A theorem concerning the invariants of linear homogeneous groups, with some applications to substitution groups. Transactions of the American Mathematical Society 5 (1904), 461-466. MR**1500684****3.**Cameron, P. J.; Kiyota, M. Sharp characters of finite groups. J. Algebra 115 (1988), 125-143. MR**937604 (89b:20026)****4.**D. Chillag, Character values of finite groups as eigenvalues of nonnegative integer matrices, Proceedings of the American Math. Society, 97 (1986), 565-567. MR**840647 (87f:20017)****5.**D. Chillag, Regular representations of semisimple algebras, separable field extensions, group characters, generalized circulants, and generalized cyclic codes, Linear Algebra and its Applications, 218 (1995), 147-183. MR**1324056 (96d:16024)****6.**I. M. Isaacs, Character Theory of Finite Groups, Academic Press, 1976. MR**0460423 (57:417)****7.**Kiyota, M. An inequality for finite permutation groups. J. Combin. Theory Ser. A 2, 1 (1979), 119. MR**541348 (81f:20009)**

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Additional Information

**David Chillag**

Affiliation:
Department of Mathematics, Technion, Israel Institute of Technology, Haifa, Israel

Email:
chillag@techunix.technion.ac.il

DOI:
https://doi.org/10.1090/S0002-9939-08-09380-5

Received by editor(s):
April 17, 2007

Published electronically:
February 14, 2008

Communicated by:
Jonathan I. Hall

Article copyright:
© Copyright 2008
American Mathematical Society