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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Counting abelian group structures
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by Francis Clarke PDF
Proc. Amer. Math. Soc. 134 (2006), 2795-2799 Request permission


A bijective proof is given of a recurrence for the function counting the number of binary operations which endow a finite set with the structure of an abelian group. The proof depends on a lemma in “labelled homological algebra” and provides a simple route to a “curious result” of Philip Hall.
  • H. Cohen and H. W. Lenstra Jr., Heuristics on class groups of number fields, Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983) Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, pp. 33–62. MR 756082, DOI 10.1007/BFb0099440
  • Euler, Leonhard, Introductio in analysin infinitorum, 1, Lausanne, 1748, Opera Omnia 8 B. G. Teubner, Geneva, 1922.
  • Hall, Philip, A partition formula connected with abelian groups, Comm. Math. Helv. 11 (1938/39), 126–129.
  • Saunders Mac Lane, Homology, Die Grundlehren der mathematischen Wissenschaften, Band 114, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0156879, DOI 10.1007/978-3-642-62029-4
  • I. G. Macdonald, The algebra of partitions, Group theory, Academic Press, London, 1984, pp. 315–333. MR 780573
  • Avinoam Mann, Philip Hall’s “rather curious” formula for abelian $p$-groups. part B, Israel J. Math. 96 (1996), no. part B, 445–448. MR 1433700, DOI 10.1007/BF02785528
  • Tomoyuki Yoshida, P. Hall’s strange formula for abelian $p$-groups, Osaka J. Math. 29 (1992), no. 3, 421–431. MR 1181111
  • Tomoyuki Yoshida, Categorical aspects of generating functions. I. Exponential formulas and Krull-Schmidt categories, J. Algebra 240 (2001), no. 1, 40–82. MR 1830543, DOI 10.1006/jabr.2000.8602
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Additional Information
  • Francis Clarke
  • Affiliation: Department of Mathematics, University of Wales Swansea, Swansea SA2 8PP, Wales
  • Email:
  • Received by editor(s): April 15, 2005
  • Published electronically: April 10, 2006
  • Communicated by: Jonathan I. Hall
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 2795-2799
  • MSC (2000): Primary 20K01; Secondary 20D60, 20K35, 20K40
  • DOI:
  • MathSciNet review: 2231600