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On Commutator Equalities and Stabilizersin Free Groups

Published online by Cambridge University Press:  20 November 2018

R. G. Burns ,
C. C. Edmunds  and
I. H. Farouqi
R. G. Burns
Affiliation:
York University, Toronto
C. C. Edmunds
Affiliation:
Mount Saint Vincent University, Halifax
I. H. Farouqi
Affiliation:
University of Karachi, Karachi
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Abstract

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A simple proof is given of a result of Hmelevskiï on the solutions of the equation [x, y] = [u, υ] over a free group for any specified u, υ. To illustrate, the equation is solved explicitly for (u, υ) = (a, b), (a2, b), ([a, b], c) (where a, b, c freely generate the free group) and thence stabilizers of the corresponding commutators in the automorphism group of this free group are determined.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 19 , Issue 3 , 01 September 1976 , pp. 263 - 267
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Appel, K. I., On two-variable equations in free groups, Proc. Amer. Math. Soc. 21 (1969), 179–185.Google Scholar
2. Coxeter, H. M. S. and Moser, W. O. J., Generators and relations for discrete groups, Springer, 1965.Google Scholar
3. Hmelevskiĭ, Ju. I., Systems of equations in a free group. I, Izv. Akad. Nauk SSSR, Ser. Mat. 35, No. 6 (1971) (A.M.S. translations: Math. USSR Izvestija 5, No. 6 (1971), 1245–1276.)Google Scholar
4. Mal’cev, A. I., On the equation zxyx -1 y -1 z -1 = -aba -1 b -1 in a free group, Algebra i Logika (Seminar) 1, No. 5 (1962), 45–50.Google Scholar