Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



Counting abelian group structures

Author: Francis Clarke
Journal: Proc. Amer. Math. Soc. 134 (2006), 2795-2799
MSC (2000): Primary 20K01; Secondary 20D60, 20K35, 20K40
Published electronically: April 10, 2006
MathSciNet review: 2231600
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: 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.

References [Enhancements On Off] (What's this?)

  • 1. Cohen, Henri and Lenstra, Hendrik W., Jr., Heuristics on class groups of number fields, 33-62, Number theory, Noordwijkerhout, 1983, Lecture Notes in Math. 1068, Berlin, 1984, Springer-Verlag. MR 756082 (85j:11144)
  • 2. Euler, Leonhard, Introductio in analysin infinitorum, 1, Lausanne, 1748, Opera Omnia 8 B. G. Teubner, Geneva, 1922.
  • 3. Hall, Philip, A partition formula connected with abelian groups, Comm. Math. Helv. 11 (1938/39), 126-129.
  • 4. Mac Lane, Saunders, Homology, Grundlehren der mathematischen Wissenschaften, 114, Springer-Verlag, Berlin, 1963.MR 0156879 (28:122)
  • 5. Macdonald, I. G., The algebra of partitions, 315-333, K. W. Gruenberg, J. Roseblade, Group theory: Essays for Philip Hall, Academic Press, London, 1984.MR 0780573 (86d:05011)
  • 6. Mann, Avinoam, Philip Hall's ``rather curious'' formula for abelian $ p$-groups, Israel J. Math. 96 (1996), 445-448. MR 1433700 (98a:20058)
  • 7. Yoshida, Tomoyuki, P. Hall's strange formula for abelian $ p$-groups, Osaka J. Math. 29 (1992), 421-431. MR 1181111 (93h:20057)
  • 8. Yoshida, Tomoyuki, Categorical aspects of generating functions. I. Exponential formulas and Krull-Schmidt categories, J. Algebra 240 (2001), 40-82. MR 1830543 (2002e:18008)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 20K01, 20D60, 20K35, 20K40

Retrieve articles in all journals with MSC (2000): 20K01, 20D60, 20K35, 20K40

Additional Information

Francis Clarke
Affiliation: Department of Mathematics, University of Wales Swansea, Swansea SA2 8PP, Wales

Received by editor(s): April 15, 2005
Published electronically: April 10, 2006
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society