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On Purifiable Subsocles of a Primary Abelian Group

Published online by Cambridge University Press:  20 November 2018

John Irwin  and
James Swanek
John Irwin
Affiliation:
Wayne State University, Detroit, Michigan
James Swanek
Affiliation:
Ford Motor Company, Dearborn, Michigan
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In this paper we shall investigate an interesting connection between the structure of G/S and G, where S is a purifiable subsocle of G. The results are interesting in the light of a counterexample by Dieudonné [3, p. 142] who exhibits a primary abelian group G, where G/S is a direct sum of cyclic groups, but G is not a direct sum of cyclic groups. Surprisingly, the assumption of the purifiability of S allows G to inherit the structure of G/S. In particular, we show that if G/S is a direct sum of cyclic groups and S supports a pure subgroup H, then G is a direct sum of cyclic groups and if is a direct summand of G which is of course a direct sum of cyclic groups. It is also shown that if G/S is a direct sum of torsion-complete groups and S supports a pure subgroup H, then G is a direct sum of torsion-complete groups and H is a direct summand of G, and is also a direct sum of torsion-complete groups.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 1 , February 1971 , pp. 48 - 57
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Crawley, P. and Jônsson, B., Refinements for infinite direct decompositions of algebraic systems, Pacific J. Math. 14 (1964), 797–855.Google Scholar
2. Cutler, D., Quasi-isomorphism for infinite abelian p-groups, Pacific J. Math. 16 (1966), 25–45.Google Scholar
3. Fuchs, L., Abelian groups (Publishing House of the Hungarian Academy of Sciences, Budapest, 1958).Google Scholar
4. Hill, P. D., The isomorphic refinement theorem for direct sums of closed groups, Proc. Amer. Math. Soc. 18 (1967), 913–919.Google Scholar
5. Hill, P. D. and Megibben, C. K., On primary groups with countable basic subgroups, Trans. Amer. Math. Soc. 124 (1966), 49–59.Google Scholar
5. Hill, P. D. and Megibben, C. K., Quasi-closed primary groups, Acta Math. Acad. Sci. Hungar. 16 (1965), 271–274.Google Scholar
7. Irwin, J. and Richman, F., Direct sums of countable groups and related concepts, J. Algebra 2 (1965), 443–450.Google Scholar
8. Irwin, J., Richman, F., and Walker, E., Countable direct sums of closed groups, Proc. Amer. Math. Soc. 17 (1966), 763–766.Google Scholar
9. Kaplansky, I., Infinite abelian groups (Univ. Michigan Press, Ann Arbor, 1954).Google Scholar
10. Koyama, T. and Irwin, J., On topological methods in abelian groups, Studies on Abelian Groups, Symposium, Montpellier, 1967, pp. 207–222 (Springer, Berlin, 1968).Google Scholar
11. Megibben, C., Large subgroups and small homomorphisms, Michigan Math. J. 13 (1966), 153–160.Google Scholar
12. Nunke, R. J., On the structure of Tor. II, Pacific J. Math. 22 (1967), 453–464.Google Scholar
13. Nunke, R. J., Homology and direct sums of countable abelian groups, Math. Z. 101 (1967), 182–212.Google Scholar
14. Nunke, R. J., Purity and subfunctors of the identity, Topics in Abelian Groups, Proc. Sympos., New Mexico State Univ., 1962, pp. 121–171 (Scott, Foresman and Co., Chicago, Illinois, 1963).Google Scholar
15. O'Neill, J., On direct products ofabelian groups, Ph.D. Dissertation, Wayne State University, Detroit, Michigan, 1967.Google Scholar
16. Pierce, R. S., Homomorphisms of primary abelian groups, Topics in Abelian Groups, Proc. Sympos., New Mexico State Univ., 1962, pp. 215–310 (Scott, Foresman and Co., Chicago, Illinois, 1963).Google Scholar
17. Richman, F., Thin abelian p-groups, Pacific J. Math. 27 (1968), 599–606.Google Scholar
18. Warfield, R., Complete abelian groups and direct sum decompositions, Ph.D. Dissertation, Harvard University, Cambridge, Massachusetts, 1967.Google Scholar