Hostname: page-component-669899f699-qzcqf Total loading time: 0 Render date: 2025-04-25T19:22:26.705Z Has data issue: false hasContentIssue false
  • English
  • Français

Direct decompositions of groups with finitely generated commutator quotient group

Part of: Special aspects of infinite or finite groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

Ronald Hirshon
Ronald Hirshon
Affiliation:
Mathematics Department Polytechnic Institute of New York333 Jay Street Brooklyn, N.Y. 11201, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let G/G' be finitely generated and let G = B1 x A1 = B2 x A2 = … = Bi x Ai = … with each Bi isomorphic to a fixed group B which obeys the maximal condition for normal subgroups. Then the Ai represent only finitely many isomorphism classes. We give an example with B infinite cyclic, G/G' free abelian of infinite (countable) rank and such that G is decomposed as above with no two Ai isomorphic.

MSC classification

Secondary: 20F06: Cancellation theory; application of van Kampen diagrams 20E22: Extensions, wreath products, and other compositions 20E15: Chains and lattices of subgroups, subnormal subgroups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 28 , Issue 3 , November 1979 , pp. 315 - 320
Copyright
Copyright © Australian Mathematical Society 1979

References

Baumslag, G. (1974), ‘Residually finite groups with the same finite images’, Compositio Math. 29, 249–252.Google Scholar
Hardy, G. H. and Wright, E. M. (1960), An Introduction to the Theory of Numbers (Clarendon Press, Oxford).Google Scholar
Hirshon, R. (1969), ‘On cancellation in groups’, Amer. Math. Monthly 76, 1037–1039.Google Scholar
Hirshon, R. (1977), ‘Some cancellation theorems with applications to nilpotent groups’, J. Austral. Math. Soc. (Ser. A) 23, 147–165.CrossRefGoogle Scholar
Hirshon, R. (1978), ‘Cancellation and Hopficity in direct products’, J. Algebra 50, 26–33.CrossRefGoogle Scholar
Hirshon, R. (1979), 'The equivalence of XtC ≈ XtD and J XC ≈ J XD, Trans. Amer. Math. Soc. 249, 331–340.Google Scholar
Pickell, P. (1974), ‘Metabelian groups with the same finite quotients’, Bull. Austral. Math. Soc. 11, 115–121.CrossRefGoogle Scholar