Hostname: page-component-77c89778f8-vsgnj Total loading time: 0 Render date: 2024-07-16T19:27:49.260Z Has data issue: false hasContentIssue false
  • English
  • Français

Commutators in Free Groups

Published online by Cambridge University Press:  20 November 2018

R. C. Lyndon  and
M. J. Wicks
R. C. Lyndon
Affiliation:
University of Michigan, Ann Arbor, Michigan 48109
M. J. Wicks
Affiliation:
University of Singapore, Singapore 1025
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the representations of an element of a free group as a commutator. For a given element g of a free group F, we are interested in the set of all pairs (x, y) of elements of F such that

(1)

where [x, y] = xyx-1 y-1. If g = 1, the problem is trivial. We assume henceforth that g ≠ 1.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 24 , Issue 1 , 01 March 1981 , pp. 101 - 106
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Burns, R. G., Edmunds, C. C, Farouqi, I. H. On commutator equalities and stabilizers in free groups. Canadian Math. Bull. 19 (1962) 263-267.Google Scholar
2. Hmelevskii, Ju. I. Systems of equations in a free group. I. Izv. Akad. Nauk SSSR, Ser. Ma. 35 (1962) 1237-1268. (Amer. Math. Soc. translation: Math. USSR Izvestija 5 (1962) 1245-1276.)Google Scholar
3. Nielsen, J. Die Isomorphismen der Allgemeinen unendlichen Gruppe mit zwei Erzeugenden. Math. Ann. 78 (1962) 385-397.Google Scholar
4. Wicks, M. J. Commutators in free products. J. London Math. Soc. 37 (1962) 433-444. Reference added in proof:Google Scholar
5. McDonough, T. P., Root-closure in free groups, J. London Math. Soc. 2 (1962) 191-192.Google Scholar