Hostname: page-component-89b8bd64d-b5k59 Total loading time: 0 Render date: 2026-05-07T00:32:00.358Z Has data issue: false hasContentIssue false
  • English
  • Français

On groups with decomposable commutator subgroups

Published online by Cambridge University Press:  18 May 2009

Robert M. Guralnick
Robert M. Guralnick
Affiliation:
Department Of Mathematics, University Of California, 405 Hilgard Avenue, Los Angeles, Ca 90024
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.

Let G be a group. We define λ(G) to be the smallest integer n such that every element of the commutator subgroup G′ is a product of n commutators. Ito [4] has shown that λ (An)= 1 for all n. Thompson [7] has shown that λ (SLn(q))= 1 for all n and q. In fact, there is no known simple group G such that λ(G)>1. However, there do exist such perfect groups (cf. [7]).

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 19 , Issue 2 , July 1978 , pp. 159 - 162
Copyright
Copyright © Glasgow Mathematical Journal Trust 1978

References

REFERENCES

1.Gallagher, P. X., The generation of the lower central series, Canad. J. Math. 17 (1965), 405–410.CrossRefGoogle Scholar
2.Guralnick, R., Expressing group elements as products of commutators, Ph.D. Thesis, UCLA (1977).Google Scholar
3.Guralnick, R., On cyclic commutator subgroups, to appear.Google Scholar
4.Ito, N., A theorem on the alternating group An (n≥5), Math. Japon. 2 (1951), 59–60.Google Scholar
5.Liebeck, H., A test for commutators, Glasgow Math. J. 17 (1976), 31–36.CrossRefGoogle Scholar
6.Macdonald, I. D., On cyclic commutator subgroups, J. London Math. Soc. 38 (1963), 419–422.CrossRefGoogle Scholar
7.Thompson, R. C., Commutators in the special and general linear groups, Trans. Amer. Math. Soc. 101, 1961, 16–33.CrossRefGoogle Scholar