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

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.**Z. Arad, J. Stavi, and M. Herzog,*Powers and products of conjugacy classes in groups*, Products of conjugacy classes in groups, Lecture Notes in Math., vol. 1112, Springer, Berlin, 1985, pp. 6–51. MR**783068**, 10.1007/BFb0072286**2.**H. F. Blichfeldt,*A theorem concerning the invariants of linear homogeneous groups, with some applications to substitution-groups*, Trans. Amer. Math. Soc.**5**(1904), no. 4, 461–466. MR**1500684**, 10.1090/S0002-9947-1904-1500684-5**3.**Peter J. Cameron and Masao Kiyota,*Sharp characters of finite groups*, J. Algebra**115**(1988), no. 1, 125–143. MR**937604**, 10.1016/0021-8693(88)90285-2**4.**David Chillag,*Character values of finite groups as eigenvalues of nonnegative integer matrices*, Proc. Amer. Math. Soc.**97**(1986), no. 3, 565–567. MR**840647**, 10.1090/S0002-9939-1986-0840647-4**5.**David Chillag,*Regular representations of semisimple algebras, separable field extensions, group characters, generalized circulants, and generalized cyclic codes*, Linear Algebra Appl.**218**(1995), 147–183. MR**1324056**, 10.1016/0024-3795(93)00167-X**6.**I. Martin Isaacs,*Character theory of finite groups*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Pure and Applied Mathematics, No. 69. MR**0460423****7.**Masao Kiyota,*An inequality for finite permutation groups*, J. Combin. Theory Ser. A**27**(1979), no. 1, 119. MR**541348**, 10.1016/0097-3165(79)90012-8

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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:
http://dx.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