Advertisement

Periodica Mathematica Hungarica

, Volume 69, Issue 1, pp 41–52 | Cite as

On the classification of abelian groups

  • Paul HillEmail author
Article
  • 91 Downloads

Remark With the exception of this remark (and perhaps one or two typos), this paper is an identical copy of a manuscript written by the author in 1967–1968 but never published in which the third axiom of countability was introduced and totally projective groups were classified. There have been dozens of subsequent papers that cited this work as “to appear” as well as in the references of a book by L. Fuchs and one by P. Griffith. To the best of the author’s knowledge, there is only one copy of the original manuscript still in existence. Taking into account these facts together with the fact that the paper is largely due to the influence of Laszlo Fuchs, it seems appropriate to publish it now in his honor.

We should advise the reader that throughout the summation symbol denotes the direct sum.

Introduction

One of the nicest structure theorems in mathematics concerns abelian groups. It is the one that classifies countable primary groups: two countable, reduced, primary groups are...

Keywords

Abelian Group Cyclic Group Projective Group Primary Group Critical Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    P. Hill, Sums of countable primary groups. Proc. Am. Math. Soc. 17, 1469–1470 (1966)zbMATHGoogle Scholar
  2. 2.
    P. Hill, C. Megibben, Extending automorphisms and lifting decompositions in abelian groups. Math. Ann. 175, 159–168 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    P. Hill, C. Megibben, On direct sums of countable groups and generalizations, in Studies on Abelian Groups (Symposium, Montpellier, 1967). (Springer, Berlin, 1968), pp. 183–206Google Scholar
  4. 4.
    I. Kaplansky, Infinite Abelian Groups (University of Michigan Press, Ann Arbor, 1954)zbMATHGoogle Scholar
  5. 5.
    I. Kaplansky, Projective modules, Ann. Math. 68(A), 371–377 (1958)MathSciNetGoogle Scholar
  6. 6.
    G. Kolettis, Direct sums of countable groups. Duke Math. J. 27, 111–125 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    R. Nunke, Purity and subfunctors of the identity, in Topics in Abelian Groups (Scott, Foresman and Co., Chicago, 1963)Google Scholar
  8. 8.
    R. Nunke, Homology and direct sums of countable groups. Math. Z. 101, 182–212 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    L.D. Parker, E.A. Walker, An extension of the Ulm-Kolettis theorems in Studies on Abelian Groups (Symposium, Montpellier, 1967). (Springer, Berlin, 1968), pp. 309–325Google Scholar
  10. 10.
    H. Ulm, Zur theorie der abzhlbar-unendlichen abelschen gruppen. Math. Ann. 107, 774–803 (1933)MathSciNetCrossRefGoogle Scholar
  11. 11.
    L. Zippin, Countable torsion groups. Ann. Math. 36, 86–99 (1935)MathSciNetCrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2014

Authors and Affiliations

  1. 1.Auburn UniversityAuburnUSA

Personalised recommendations