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Commutators of Certain Finitely Generated Soluble Groups

Published online by Cambridge University Press:  20 November 2018

A. H. Rhemtulla
A. H. Rhemtulla*
Affiliation:
University of Alberta, Edmonton, Alberta
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Groups in which the commutator subgroup coincides with the set of commutators have been studied to a certain extent by several authors. It was shown in (2; 4; 6; 7) that various types of known simple groups have this property. In (3), Macdonald has considered certain soluble groups with this property, and Hall has shown that any group can be embedded as a subgroup of a simple group of this type. Here we shall be concerned with the class C of groups defined as follows.

For any positive integer n, denote by Cn the class of all groups in which every element of the commutator subgroup can be expressed as a product of at most n commutators. It is not difficult to show that Cn is a proper subclass of Cn+1 for all n. Let so that a group G ∊ C if and only if G ∊ Cn for some n.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1160 - 1164
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Hall, P., Finiteness condition for soluble groups, Proc. London Math. Soc. (3) 4 (1954), 419–436.Google Scholar
2. Itô, N., A theorem on the alternating group Un„ (n ≧ 5), Math. Japon. 2 (1951), 59–60 Google Scholar
3. Macdonald, I. D., On cyclic commutator subgroups, J. London Math. Soc. 88 (1963), 419–422.Google Scholar
4. Ore, O., Some remarks on commutators, Proc. Amer. Math. Soc. 2 (1951), 307–314.Google Scholar
5. Rhemtulla, A. H., Problems of bounded expressibility in free products, Proc, Cambridge Philos. Soc. 64 (1968), 573–584.Google Scholar
6. Thompson, R. C., Commutators in the special and general linear groups, Trans. Amer. Math. Soc. 101 (1961), 16–33.Google Scholar
8. Villari, G., Sui Commutatori del gruppo modulare, Bull. Un. Mat. Ital. (3) 13 (1958), 196–201.Google Scholar