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Systems of equations and generalized characters in groups

Published online by Cambridge University Press:  20 November 2018

I. M. Isaacs
I. M. Isaacs*
Affiliation:
University of Wisconsin, Madison, Wisconsin
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Let F be the free group on n generators x1, …, Xn and let G be an arbitrary group. An element ω ∈ F determines a function x → ω(x) from n-tuples x = (x1, x2, …, xn) ∈ Gn into G. In a recent paper [5] Solomon showed that if ω1, ω2, …, ωm ∈ F with m < n, and K1, …, Km are conjugacy classes of a finite group G, then the number of x ∈ Gn with ωi(x) ∈ Ki for each i, is divisible by |G|. Solomon proved this by constructing a suitable equivalence relation on Gn.

Another recent application of an unusual equivalence relation in group theory is in Brauer's paper [1], where he gives an elementary proof of the Frobenius theorem on solutions of xk = 1 in a group.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 5 , 01 October 1970 , pp. 1040 - 1046
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Brauer, R., On a theorem of Frobenius, Amer. Math. Monthly 76 (1969), 12–15.Google Scholar
2. Coxeter, H. S. M. and Moser, W. O., Generators and relations for discrete groups, Second Ed. (Springer-Verlag, New York, 1965).Google Scholar
3. Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras (Interscience, New York, 1962).Google Scholar
4. Magnus, W., Karrass, A., and Solitar, D., Combinational group theory (Interscience, New York, 1966).Google Scholar
5. Solomon, L., The solution of equations in groups, Arch. Math. 20 (1969), 241–247.Google Scholar