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The freeness of some projective metabelian groups

Published online by Cambridge University Press:  17 April 2009

A.J. McIsaac
Affiliation:
Mathematical Institute, University of Oxford, Oxford, England.
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Abstract

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The question whether there exist non-free projective groups of rank r in the variety has been answered in the affirmative for n ≥ 2, r ≥ 2, except for n = r = 2, by V.A. Artamonov. This paper consists in a proof that a projective group G of rank 2 in is free. If x and y are any two elements which generate G modulo , then the group F generated by x and y is free in , and the index of F in G is finite and not divisible by 2. One wishes to replace x by xu and y by , where u and lie in , so that 〈xu, yν〉 is the whole of G. This can be done: first, on general grounds, it is sufficient that 〈xu, yν〉 contain every C(a), where C(a) is the centralizer in the G/-module of an element a in (and moreover choices of u and ν for each C(a) can be combined to give a single choice good for all C(a)); second, for the particular small numbers involved, the structure of C(a) is sufficiently simple for one to pick suitable u and ν without trouble.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1975

References

[1]Artamonov, V.A., “Projective metabelian nonfree groups”, Bull. Austral. Math. Soc. 13 (1975), 101115.CrossRefGoogle Scholar
[2]Hall, P., “The splitting properties of relatively free groups”, Proc. London Math. Soc. (3) 4 (1954), 343356.CrossRefGoogle Scholar