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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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3. Clifford, A. H., Representations induced in an invariant subgroup, Ann. of Math. 38 (1937), 533–50.Google Scholar
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