## A classification theorem for abelian $p$-groups

HTML articles powered by AMS MathViewer

- by R. B. Warfield
- Trans. Amer. Math. Soc.
**210**(1975), 149-168 - DOI: https://doi.org/10.1090/S0002-9947-1975-0372071-2
- PDF | Request permission

## Abstract:

A new class of Abelian $p$-groups, called $S$-groups, is studied, and the groups in this class are classified in terms of cardinal invariants. The class of $S$-groups includes Nunke’s totally projective $p$-groups. The invariants consist of the Ulm invariants (which Hill has shown can be used to classify the totally projective groups) together with a new sequence of invariants indexed by limit ordinals which are not cofinal with $\omega$. The paper includes a fairly complete discussion of dense isotype subgroups of totally projective $p$-groups, including necessary and sufficient conditions for two of them to be congruent under the action of an automorphism of the group. It also includes an extension of Ulm’s theorem to a class of mixed modules over a discrete valuation ring.## References

- B. Edington,
- L. Fuchs,
*Notes on abelian groups. II*, Acta Math. Acad. Sci. Hungar.**11**(1960), 117–125 (English, with Russian summary). MR**125878**, DOI 10.1007/BF02020629 - László Fuchs,
*Infinite abelian groups. Vol. II*, Pure and Applied Mathematics. Vol. 36-II, Academic Press, New York-London, 1973. MR**0349869** - D. K. Harrison,
*Infinite abelian groups and homological methods*, Ann. of Math. (2)**69**(1959), 366–391. MR**104728**, DOI 10.2307/1970188 - D. K. Harrison,
*On the structure of $\textrm {Ext}$*, Topics in Abelian Groups (Proc. Sympos., New Mexico State Univ., 1962) Scott, Foresman & Co., Chicago, Ill., 1963, pp. 195–209. MR**0242926** - Paul Hill,
*Isotype subgroups of direct sums of countable groups*, Illinois J. Math.**13**(1969), 281–290. MR**240198**
P. D. Hill, - 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** - John M. Irwin and Elbert A. Walker,
*On $N$-high subgroups of Abelian groups*, Pacific J. Math.**11**(1961), 1363–1374. MR**136653**, DOI 10.2140/pjm.1961.11.1363 - J. M. Irwin and E. A. Walker,
*On isotype subgroups of Abelian groups*, Bull. Soc. Math. France**89**(1961), 451–460. MR**147539**, DOI 10.24033/bsmf.1570 - Irving Kaplansky,
*Infinite abelian groups*, Revised edition, University of Michigan Press, Ann Arbor, Mich., 1969. MR**0233887** - L. Ya. Kulikov,
*Generalized primary groups. I*, Trudy Moskov. Mat. Obšč.**1**(1952), 247–326 (Russian). MR**0049188** - Charles Megibben,
*On $p^{\alpha }$-high injectives*, Math. Z.**122**(1971), 104–110. MR**299678**, DOI 10.1007/BF01110084 - R. J. Nunke,
*Modules of extensions over Dedekind rings*, Illinois J. Math.**3**(1959), 222–241. MR**102538**, DOI 10.1215/ijm/1255455124 - R. J. Nunke,
*Purity and subfunctors of the identity*, Topics in Abelian Groups (Proc. Sympos., New Mexico State Univ., 1962) Scott, Foresman & Co., Chicago, Ill., 1963, pp. 121–171. MR**0169913** - R. J. Nunke,
*Homology and direct sums of countable abelian groups*, Math. Z.**101**(1967), 182–212. MR**218452**, DOI 10.1007/BF01135839 - R. J. Nunke,
*On the structure of $\textrm {Tor}$. II*, Pacific J. Math.**22**(1967), 453–464. MR**214659**, DOI 10.2140/pjm.1967.22.453 - R. J. Nunke,
*Uniquely elongating modules*, Symposia Mathematica, Vol. XIII (Convegno di Gruppi Abeliani & Convegno di Gruppi e loro Rappresentazioni, INDAM, Rome, 1972) Academic Press, London, 1974, pp. 315–330. MR**0364491** - 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**
R. S. Pierce, - 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 - R. B. Warfield Jr.,
*Classification theorems for $p$-groups and modules over a discrete valuation ring*, Bull. Amer. Math. Soc.**78**(1972), 88–92. MR**291284**, DOI 10.1090/S0002-9904-1972-12870-2
B. Wick,

*Isomorphism invariants in quotient categories of Abelian groups*, Thesis, New Mexico State University, 1972.

*On the classification of Abelian groups*, Mimeographed notes, 1968.

*Abelian Group Seminar Notes*, New Mexico State University, 1968.

*Classification theorems for infinite Abelian groups*, Thesis, University of Washington, 1972.

## Bibliographic Information

- © Copyright 1975 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**210**(1975), 149-168 - MSC: Primary 20K10
- DOI: https://doi.org/10.1090/S0002-9947-1975-0372071-2
- MathSciNet review: 0372071