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Proceedings of the American Mathematical Society
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
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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.


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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: http://dx.doi.org/10.1090/S0002-9939-03-06920-X
PII: S 0002-9939(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