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)

 

 

The class equation and counting in factorizable monoids


Authors: S. Lipscomb and J. Konieczny
Journal: Proc. Amer. Math. Soc. 131 (2003), 3345-3351
MSC (2000): Primary 20M20, 20M15
Published electronically: February 14, 2003
MathSciNet review: 1990622
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For orders and conjugacy in finite group theory, Lagrange's Theorem and the class equation have universal application. Here, the class equation (extended to monoids via standard group action by conjugation) is applied to factorizable submonoids of the symmetric inverse monoid. In particular, if $M$is a monoid induced by a subgroup $G$ of the symmetric group $S_n$, then the center $Z_{\makebox{\tiny$G$ }}(M)$ (all elements of $M$ that commute with every element of $G$) is $Z(G) \cup\{0\}$ if and only if $G$ is transitive. In the case where $G$ is both transitive and of order either $p$ or $p^2$ (for $p$prime), formulas are provided for the order of $M$ as well as the number and sizes of its conjugacy classes.


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

  • 1. S. Y. Chen and S. C. Hsieh, Factorizable inverse semigroups, Semigroup Forum 8 (1974), no. 4, 283–297. MR 0376914
  • 2. John M. Howie, Fundamentals of semigroup theory, London Mathematical Society Monographs. New Series, vol. 12, The Clarendon Press, Oxford University Press, New York, 1995. Oxford Science Publications. MR 1455373
  • 3. Nathan Jacobson, Basic algebra. I, 2nd ed., W. H. Freeman and Company, New York, 1985. MR 780184
  • 4. Mark V. Lawson, Inverse semigroups, World Scientific Publishing Co., Inc., River Edge, NJ, 1998. The theory of partial symmetries. MR 1694900
  • 5. S. L. Lipscomb, Problems and applications of finite inverse semigroups, Words, languages and combinatorics (Kyoto, 1990) World Sci. Publ., River Edge, NJ, 1992, pp. 337–352. MR 1161033
  • 6. Stephen Lipscomb, Symmetric inverse semigroups, Mathematical Surveys and Monographs, vol. 46, American Mathematical Society, Providence, RI, 1996. MR 1413301
  • 7. Yupaporn Tirasupa, Factorizable transformation semigroups, Semigroup Forum 18 (1979), no. 1, 15–19. MR 537659, 10.1007/BF02574171
  • 8. Helmut Wielandt, Finite permutation groups, Translated from the German by R. Bercov, Academic Press, New York-London, 1964. MR 0183775

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 20M20, 20M15

Retrieve articles in all journals with MSC (2000): 20M20, 20M15


Additional Information

S. Lipscomb
Affiliation: Department of Mathematics, Mary Washington College, Fredericksburg, Virginia 22401
Email: slipscom@mwc.edu

J. Konieczny
Affiliation: Department of Mathematics, Mary Washington College, Fredericksburg, Virginia 22401
Email: jkoniecz@mwc.edu

DOI: https://doi.org/10.1090/S0002-9939-03-06920-X
Keywords: Factorizable monoids, symmetric inverse semigroups, class equation, conjugacy classes, permutation groups, transformation semigroups
Received by editor(s): April 12, 2002
Received by editor(s) in revised form: June 5, 2002
Published electronically: February 14, 2003
Communicated by: Stephen D. Smith
Article copyright: © Copyright 2003 American Mathematical Society