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Counting abelian group structures
Author:
Francis Clarke
Journal:
Proc. Amer. Math. Soc. 134 (2006), 2795-2799
MSC (2000):
Primary 20K01; Secondary 20D60, 20K35, 20K40
Posted:
April 10, 2006
MathSciNet review:
2231600
Full-text PDF Free Access
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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
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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.
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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
 -groups, Israel J. Math. 96 (1996), 445-448. MR 1433700 (98a:20058)
- 7.
Yoshida, Tomoyuki, P. Hall's strange formula for abelian
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Yoshida, Tomoyuki, Categorical aspects of generating functions. I. Exponential formulas and Krull-Schmidt categories, J. Algebra 240 (2001), 40-82. MR 1830543 (2002e:18008)
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Additional Information
Francis Clarke
Affiliation:
Department of Mathematics, University of Wales Swansea, Swansea SA2 8PP, Wales
Email:
F.Clarke@Swansea.ac.uk
DOI:
http://dx.doi.org/10.1090/S0002-9939-06-08396-1
PII:
S 0002-9939(06)08396-1
Received by editor(s):
April 15, 2005
Posted:
April 10, 2006
Communicated by:
Jonathan I. Hall
Article copyright:
© Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain after
28 years from publication.
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