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On a set of normal subgroups

Published online by Cambridge University Press:  18 May 2009

I. D. Macdonald
I. D. Macdonald
Affiliation:
The University Sheffield, 10
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The commutator [a, b] of two elements a and b in a group G satisfies the identity

ab = ba[a, b].

The subgroups we study are contained in the commutator subgroup G′, which is the subgroup generated by all the commutators.

The group G is covered by a well-known set of normal subgroups, namely the normal closures {g}G of the cyclic subgroups {g} in G. In a similar way one may associate a subgroup K(g) with each element g, by defining K(g) to be the subgroup generated by the commutators [g, x] as x takes all values in G. These subgroups generate G′ (but do not cover G′ in general), and are normal in G in consequence of the identical relation

(A) [g, x]Y = [g, y]−1[g, xy]

holding for all g, x and y in G. (By ab we mean b−1ab.) It is easy to see that

{g}G = {g, K(g)}.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 5 , Issue 3 , January 1962 , pp. 137 - 146
Copyright
Copyright © Glasgow Mathematical Journal Trust 1962

References

1.Macdonald, I. D., On certain varieties of groups, Math. Z. 76 (1961), 270–282.CrossRefGoogle Scholar
2.Neumann, B. H., Groups with finite classes of conjugate elements, Proc. London Math. Soc. (3) 1 (1951), 178–187.Google Scholar
3.Neumann, B. H., Groups covered by permutable subsets, J. London Math. Soc. 29 (1954), 236–248.CrossRefGoogle Scholar
4.Ore, O., Some remarks on commutators, Proc. Amer. Math. Soc. 2 (1951), 307–314.CrossRefGoogle Scholar
5.Zassenhaus, H. J., The theory of groups (New York, 1958).Google Scholar