Geodesic
Systems of equations and generalized characters in groups
Canadian journal of mathematics, Tome 22 (1970) no. 5, pp. 1040-1046

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Let F be the free group on n generators x 1, ..., Xn and let G be an arbitrary group. An element ω ∈ F determines a function x → ω(x) from n-tuples x = (x 1, x 2, ..., xn ) ∈ Gn into G. In a recent paper [5] Solomon showed that if ω 1, ω 2, ..., ωm ∈ F with m < n, and K 1, ..., 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. 
Isaacs, I. M. Systems of equations and generalized characters in groups. Canadian journal of mathematics, Tome 22 (1970) no. 5, pp. 1040-1046. doi: 10.4153/CJM-1970-120-0
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[1] 1. Brauer, R., On a theorem of Frobenius, Amer. Math. Monthly 76 (1969), 12–15. Google Scholar

[2] 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] 3. Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras (Interscience, New York, 1962). Google Scholar

[4] 4. Magnus, W., Karrass, A., and Solitar, D., Combinational group theory (Interscience, New York, 1966). Google Scholar

[5] 5. Solomon, L., The solution of equations in groups, Arch. Math. 20 (1969), 241–247. Google Scholar

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