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Products of Commutators as Products of Squares

Published online by Cambridge University Press:  20 November 2018

Charles C. Edmunds
Charles C. Edmunds*
Affiliation:
University of Manitoba, Winnipeg, Manitoba
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In any group G, the commutator subgroup G' is contained in G2, the subgroup of G generated by the squares in G. Thus any product of commutators can be written as a product of squares in G. For instance, the commutator [x, y] ( = xyx-1y-1) can be expressed as the product of three squares: [x, y] = x2(x-1y)2(y-1)2. Roger Lyndon and Morris Newman have made the interesting observation [4, Theorem 1] that, in this case, the number three is minimal in the sense that there are groups which contain commutators not expressible as the product of fewer than three squares.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 6 , 01 December 1975 , pp. 1329 - 1335
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Dehn, M., Ùber unendliche diskontinuierliche Gruppen, Math. Ann. 71 (1912), 116–144.Google Scholar
2. Edmunds, C., On the endomorphism problem for free groups, Comm. Algebra 3 (1975), 1–20.Google Scholar
3. Edmunds, C., Some properties of quadratic words in free groups (to appear in Proc. Amer. Math. Soc.).Google Scholar
4. Lyndon, R. and Newman, M., Commutators as products of squares, Proc. Amer. Math. Soc. S9 (1973), 267–272.Google Scholar
5. Magnus, W., Karrass, A., and Solitar, D., Combinatorial group theory (Interscience, New York, 1966).Google Scholar