The third axiom of countability for abelian groups

Author:
Paul Hill

Journal:
Proc. Amer. Math. Soc. **82** (1981), 347-350

MSC:
Primary 20K10; Secondary 20K15, 20K20

MathSciNet review:
612716

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Abstract | References | Similar Articles | Additional Information

Abstract: Three different definitions of the third axiom of countability for abelian -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.

**[1]**Peter Crawley and Alfred W. Hales,*The structure of torsion abelian groups given by presentations*, Bull. Amer. Math. Soc.**74**(1968), 954–956. MR**0232840**, 10.1090/S0002-9904-1968-12102-0**[2]**László Fuchs,*Infinite abelian groups. Vol. II*, Academic Press, New York-London, 1973. Pure and Applied Mathematics. Vol. 36-II. MR**0349869****[3]**Phillip A. Griffith,*Infinite abelian group theory*, The University of Chicago Press, Chicago, Ill.-London, 1970. MR**0289638****[4]**P. Hill,*On the classification of abelian groups*, xeroxed manuscript, 1967.**[5]**Paul Hill,*A countability condition for primary groups presented by relations of length two*, Bull. Amer. Math. Soc.**75**(1969), 780–782. MR**0246959**, 10.1090/S0002-9904-1969-12287-1**[6]**Paul Hill,*On the decomposition of certain infinite nilpotent groups*, Math. Z.**113**(1970), 237–248. MR**0286888****[7]**R. J. Nunke,*Homology and direct sums of countable abelian groups*, Math. Z.**101**(1967), 182–212. MR**0218452****[8]**Elbert A. Walker,*Ulm’s theorem for totally projective groups*, Proc. Amer. Math. Soc.**37**(1973), 387–392. MR**0311805**, 10.1090/S0002-9939-1973-0311805-3

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1981-0612716-0

Keywords:
Third axiom of countability,
totally projective group,
ascending chain of nice subgroups,
Ulm invariant,
torsionfree group,
pure subgroup

Article copyright:
© Copyright 1981
American Mathematical Society