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Replacement of Factors By Subgroups in the Factorization of Abelian Groups
Author(s) -
Sands A. D.
Publication year - 2000
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609300007189
Subject(s) - mathematics , abelian group , factorization , cyclic group , prime (order theory) , elementary abelian group , prime factor , order (exchange) , combinatorics , integer (computer science) , group (periodic table) , product (mathematics) , free abelian group , rank of an abelian group , pure mathematics , algorithm , chemistry , geometry , organic chemistry , finance , computer science , economics , programming language
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.