The third axiom of countability for abelian groups
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- by Paul Hill PDF
- Proc. Amer. Math. Soc. 82 (1981), 347-350 Request permission
Abstract:
Three different definitions of the third axiom of countability for abelian $p$-groups are shown to be equivalent. The main interest in this stems from the fact that the third axiom of countability characterizes one of the most important classes of abelian groups. Moreover, the equivalence of two of these definitions validates the proof of Theorem 67 in P. Griffith’s Infinite abelian group theory. As a further application of our method of proof, we show that every torsionfree abelian group satisfies the third axiom of countability with respect to purity.References
- Peter Crawley and Alfred W. Hales, The structure of torsion abelian groups given by presentations, Bull. Amer. Math. Soc. 74 (1968), 954–956. MR 232840, DOI 10.1090/S0002-9904-1968-12102-0
- László Fuchs, Infinite abelian groups. Vol. II, Pure and Applied Mathematics. Vol. 36-II, Academic Press, New York-London, 1973. MR 0349869
- Phillip A. Griffith, Infinite abelian group theory, University of Chicago Press, Chicago, Ill.-London, 1970. MR 0289638 P. Hill, On the classification of abelian groups, xeroxed manuscript, 1967.
- Paul Hill, A countability condition for primary groups presented by relations of length two, Bull. Amer. Math. Soc. 75 (1969), 780–782. MR 246959, DOI 10.1090/S0002-9904-1969-12287-1
- Paul Hill, On the decomposition of certain infinite nilpotent groups, Math. Z. 113 (1970), 237–248. MR 286888, DOI 10.1007/BF01110196
- R. J. Nunke, Homology and direct sums of countable abelian groups, Math. Z. 101 (1967), 182–212. MR 218452, DOI 10.1007/BF01135839
- Elbert A. Walker, Ulm’s theorem for totally projective groups, Proc. Amer. Math. Soc. 37 (1973), 387–392. MR 311805, DOI 10.1090/S0002-9939-1973-0311805-3
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 82 (1981), 347-350
- MSC: Primary 20K10; Secondary 20K15, 20K20
- DOI: https://doi.org/10.1090/S0002-9939-1981-0612716-0
- MathSciNet review: 612716