On the structure of abelian 𝑝-groups

Hill, Paul

doi:10.1090/S0002-9947-1985-0776390-3

On the structure of abelian $p$-groups
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Trans. Amer. Math. Soc. 288 (1985), 505-525 Request permission

Abstract:

A new kind of abelian $p$-group, called an $A$-group, is introduced. This class contains the totally projective groups and Warfield’s $S$-groups as special cases. It also contains the $N$-groups recently classified by the author. These more general groups are classified by cardinal (numerical) invariants which include, but are not limited to, the Ulm-Kaplansky invariants. Thus the existing theory, as well as the classification, of certain abelian $p$-groups is once again generalized. Having classified $A$-groups (by means of a uniqueness and corresponding existence theorem) we can successfully study their structure and special properties. Such a study is initiated in the last section of the paper.

References

Peter Crawley and Alfred W. Hales, The structure of abelian $p$-groups given by certain presentations, J. Algebra 12 (1969), 10–23. MR 238947, DOI 10.1016/0021-8693(69)90014-3
László Fuchs, Infinite abelian groups. Vol. II, Pure and Applied Mathematics. Vol. 36-II, Academic Press, New York-London, 1973. MR 0349869
Paul Hill, Isotype subgroups of direct sums of countable groups, Illinois J. Math. 13 (1969), 281–290. MR 240198
Paul Hill, The third axiom of countability for abelian groups, Proc. Amer. Math. Soc. 82 (1981), no. 3, 347–350. MR 612716, DOI 10.1090/S0002-9939-1981-0612716-0

On the classification of abelian groups

Paul Hill, The classification of $N$-groups, Houston J. Math. 10 (1984), no. 1, 43–55. MR 736574
Paul Hill and Charles Megibben, On direct sums of countable groups and generalizations, Studies on Abelian Groups (Symposium, Montpellier, 1967) Springer, Berlin, 1968, pp. 183–206. MR 0242943
Paul Hill and Charles Megibben, Extending automorphisms and lifting decompositions in Abelian groups, Math. Ann. 175 (1968), 159–168. MR 223449, DOI 10.1007/BF01418771

On the theory and classification of abelian

groups

George Kolettis Jr., Direct sums of countable groups, Duke Math. J. 27 (1960), 111–125. MR 110748
Charles Megibben, On $p^{\alpha }$-high injectives, Math. Z. 122 (1971), 104–110. MR 299678, DOI 10.1007/BF01110084
R. J. Nunke, On the structure of $\textrm {Tor}$, Proc. Colloq. Abelian Groups (Tihany, 1963) Akadémiai Kiadó, Budapest, 1964, pp. 115–124. MR 0169912
R. J. Nunke, Homology and direct sums of countable abelian groups, Math. Z. 101 (1967), 182–212. MR 218452, DOI 10.1007/BF01135839
Roger Hunter, Fred Richman, and Elbert Walker, Warfield modules, Abelian group theory (Proc. Second New Mexico State Univ. Conf., Las Cruces, N.M., 1976) Lecture Notes in Math., Vol. 616, Springer, Berlin, 1977, pp. 87–123. MR 0506216
Larry D. Parker and Elbert A. Walker, An extension of the Ulm-Kolettis theorems, Studies on Abelian Groups (Symposium, Montpellier, 1967) Springer, Berlin, 1968, pp. 309–325. MR 0265458
Helmut Ulm, Zur Theorie der abzählbar-unendlichen Abelschen Gruppen, Math. Ann. 107 (1933), no. 1, 774–803 (German). MR 1512826, DOI 10.1007/BF01448919
Kyle D. Wallace, On mixed groups of torsion-free rank one with totally projective primary components, J. Algebra 17 (1971), 482–488. MR 272891, DOI 10.1016/0021-8693(71)90005-6
R. B. Warfield Jr., A classification theorem for abelian $p$-groups, Trans. Amer. Math. Soc. 210 (1975), 149–168. MR 372071, DOI 10.1090/S0002-9947-1975-0372071-2
Robert B. Warfield Jr., The structure of mixed abelian groups, Abelian group theory (Proc. Second New Mexico State Univ. Conf., Las Cruces, N.M., 1976) Lecture Notes in Math., Vol. 616, Springer, Berlin, 1977, pp. 1–38. MR 0507073
Leo Zippin, Countable torsion groups, Ann. of Math. (2) 36 (1935), no. 1, 86–99. MR 1503210, DOI 10.2307/1968666