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On the Decomposition of Groups

Published online by Cambridge University Press:  20 November 2018

Paul Hill
Paul Hill*
Affiliation:
University of Houston, Houston, Texas
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The problem in which we are interested is the following. Call an additively written group G finitely decomposable if G = Σ Gi is the weak sum of finite groups Gi, Consider the following property.

Property P. Each subgroup of G having cardinality less than G is contained in a finitely decomposable direct summand of G.

Does Property P imply that G is finitely decomposable? We shall demonstrate that the answer is negative even in the commutative case. Our question is closely related to (1, Problem 5). In (4), an abelian group is called a Fuchs 5-group if every infinite subgroup of the group can be embedded in a direct summand of the same cardinality. The question of whether or not a Fuchs 5-group is in fact a direct sum of countable groups has been open for several years.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 762 - 768
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Fuchs, L., Abelian groups (Publishing House of the Hungarian Academy of Sciences, Budapest, 1958).Google Scholar
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3. Hill, P. and Megibben, C., Extending automorphisms and lifting decompositions in abelian groups, Math. Ann. 175 (1968), 159–168.Google Scholar
4. Irwin, J. and Richman, F., Direct sums of countable groups and related concepts, J. Algebra 2 (1965), 443–450.Google Scholar
5. Kaplansky, I., Projective modules, Ann. of Math. (2) 68 (1958), 371–377.Google Scholar