The class equation and counting in factorizable monoids

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 $Z_G(M)$ (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.