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Bulletin of the London Mathematical Society 2000 32(3):297-304; doi:10.1112/S0024609300007189
© 2000 by London Mathematical Society
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© London Mathematical Society

Replacement of Factors By Subgroups in the Factorization of Abelian Groups

A. D. Sands

Department of Mathematics, Dundee University Dundee DD1 4HN

Received 27 February 1999. Revision received 15 September 1999.

In his book Abelian groups, L. Fuchs raised the question as to whether, in general, in the factorization of a finite (cyclic) abelian group one factor may always be replaced by some subgroup. The answer turned out to be negative in general, but positive in certain cases. In this paper the complete answer for cyclic groups is given. In all previously unsolved cases, the answer turns out to be positive. It is shown that a cyclic group has the property that in every factorization, one factor may be replaced by a subgroup if and only if the group has order equal to the product of a prime and a square-free integer. Certain results are also given in non-cyclic cases. 1991 Mathematics Subject Classification 20K01.


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