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Further Duals of a verbal Subgroup

Published online by Cambridge University Press:  18 May 2009

S. Moran
S. Moran
Affiliation:
The University, Glasgow
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In a previous paper [3] we gave two methods for constructing subgroups which in certain senses may be considered to be dual to a verbal subgroup Vf(G) of an arbitrary group G. Associated with a word h (u, v) in the two symbols u and v, we have (i) the first dual subgroup which is defined as the minimal subgroup of G containing all elements ξ of G for which

for all values of x1, x2, …, in xn in G, and (ii), the second dual subgroup which is defined as the minimal subgroup of G containing all elements z of G for which

for all values of x1, x2, …, xn in G. Below we introduce slight variations to these definitions, which give rise to the concepts of the third and the fourth dual subgroups respectively. For certain values of h(u, v) we obtain concepts which also arise from and , namely, the marginal subgroup, the invariable subgroup and the centralizer of a verbal subgroup. We also obtain the new concepts of elemental subgroups and commutal subgroups and briefly sketch some of their properties. Finally we conclude by showing that MacLane's dual for the centralizer of a verbal subgroup is a closely related verbal subgroup.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 4 , Issue 2 , July 1959 , pp. 88 - 91
Copyright
Copyright © Glasgow Mathematical Journal Trust 1959

References

1.MacLane, S., Groups, categories and duality, Proc. Nat. Acad. Sci., 34 (1948), 263–267.Google Scholar
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3.Moran, S., Duals of a verbal subgroup, J. London Math. Soc., 33 (1958), 220–236; Corrigenda, 34(1959), 250.CrossRefGoogle Scholar
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