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The Generation of the Lower Central Series

Published online by Cambridge University Press:  20 November 2018

P. X. Gallagher
P. X. Gallagher*
Affiliation:
Columbia University, New York
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Let G be a finite group with commutator subgroup G′. In an earlier paper (4) it was shown that each element of G′ is a product of n commutators, if 4n ≥ |G′|. The object of this paper is to improve this result in two directions:

Theorem 1a. If (n + 2)!n! > 2|G′| — 2, then each element of G′ is a product of n commutators.

Theorem 1b. If G is a p-group, with |G′| = pa, and if n(n + 1) > a, then each element of G′ is a product of n commutators.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 17 , 1965 , pp. 405 - 410
Copyright
Copyright © Canadian Mathematical Society 1965

References

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4. Gallagher, P. X., Group characters and commutators, Math. Z., 79 (1962), 122–6.Google Scholar
5. Macdonald, I. D., On a set of normal subgroups, Proc. Glasgow Math. Assoc, 5 (1962), 137–46.Google Scholar