Replacement of Factors By Subgroups in the Factorization of Abelian Groups

In his book Abelian groups, L. Fuchs raised the question asto 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 positivein certain cases. In this paper the complete answer for cyclicgroups is given. In all previously unsolved cases, the answerturns out to be positive. It is shown that a cyclic group hasthe property that in every factorization, one factor may bereplaced by a subgroup if and only if the group has order equalto the product of a prime and a square-free integer. Certainresults are also given in non-cyclic cases. 1991 MathematicsSubject Classification 20K01.