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Nilpotent Extensions of Abelian p-GROUPS

Published online by Cambridge University Press:  20 November 2018

Joseph Buckley  and
James Wiegold
Joseph Buckley
Affiliation:
Western Michigan University, Kalamazoo, Michigan
James Wiegold
Affiliation:
University College, Cardiff, Wales
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This paper arose out of an attempt to solve the following problem due to Suprunenko [5, Problem 2.77]. For which pairs of abelian groups A, B is every extension of A by B nilpotent? We obtain complete answers when A and B are p-groups and (a) A has finite exponent or (b) B is divisible or (c) A has infinite exponent, is countable and B is non-divisible. The structure of a basic subgroup of A plays a central role in cases (b) and (c).

At the outset we must say that the problem is too difficult to solve in complete generality. If G/A ≅ 2?, then the nilpotency of G depends solely on properties of the associated homomorphism θ.B → Aut A. Thus for instance if A is torsion-free and B finite, G is nilpotent if and only if the extension is a central one, and we would need detailed information on finite subgroups of the group Aut A.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 38 , Issue 5 , 01 October 1986 , pp. 1025 - 1052
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Baumslag, G., Wreath products and p-groups, Proc. Cambridge Philos. Soc. 55 (1959), 224–231.Google Scholar
2. de Groot, J., Indecomposable abelian groups, Nederl. Acad. Wetensch. Proc. Ser. A 60 (1957), 137–145.Google Scholar
3. Fuchs, L., Abelian groups (Budapest, 1958).Google Scholar
4. Hartley, B., A dual approach to Černikov modules, Proc. Cambridge Philos. Soc. 82 (1977), 215–239.Google Scholar
5. Kourovka Notebook [Unsolvedproblems in group theory], (Novosibirsko, 1981).Google Scholar
6. Liebeck, H., Concerning nilpotent wreath products, Proc. Cambridge Philos. Soc. 58 (1962), 443–451.Google Scholar
7. Zaleskii, A. E., A nilpotent p-group has an outer automorphism, Dokl. Akad. Nauk. SSSR 196 (1971).Google Scholar