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A note on generalized characters

Published online by Cambridge University Press:  17 April 2009

R.M. Bryant  and
L.G. Kovács
R.M. Bryant
Affiliation:
Institute of Advanced Studies, Australian National University, Canberra, ACT.
L.G. Kovács
Affiliation:
Institute of Advanced Studies, Australian National University, Canberra, ACT.
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Abstract

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Let α be a generalized character of a (not necessarily finite) group G over an arbitrary field, and n a positive integer. It is shown that the function α(n) defined by α(n) (g) = α(gn) is also a generalized character of G. An application confirms a conjecture of Robert Higgins and David Ballew: if G is finite and k is also a positive integer, the k-th power of the number of n-th roots of g, summed over all g in G, is divisible by the order of G.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 5 , Issue 2 , October 1971 , pp. 265 - 269
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Boerner, Hermann, Representations of groups with special consideration for the needs of modern physics (translated by Murphy, P.G. in cooperation with Mayer-Kalkschmidt, J. and Care, P.. North-Holland, Amsterdam; Interscience [John Wiley & Sons], New York, 1963).Google Scholar
[2]Curtis, Charles W.; Reiner, Irving, Representation theory of finite groups and associative algebras (Pure and Applied Mathematics, XI. Interscience [John Wiley & Sons], New York, London, 1962).Google Scholar
[3]Higgins, Robert and Ballew, David, “An equation for finite groups”, Amer. Math. Monthly 78 (1971), 274–275.CrossRefGoogle Scholar
[4]Rudvalis, A. and Snapper, E., “Numerical polynomials for arbitrary characters”, J. Combinatorial Theory Ser. A 10 (1971), 145–159.CrossRefGoogle Scholar
[5]Waerden, B.L. van der, Modern algebra, Vol. II (translated from the second revised German edition by Benac, Theodore J.. Frederick Ungar, New York, 1950).Google Scholar