Hultman Numbers and Generalized Commuting Probability in Finite Groups

Abstract

Let $G$ be a finite group and $\pi$ be a permutation from $S_n$. We investigate the distribution of the probabilities of the equality \[ a_{1}a_{2}\cdots a_{n-1}a_{n}=a_{π_{1}}a_{π_{2}}\cdots a_{π_{n-1}}a_{π_{n}} \] when $\pi$ varies over all the permutations in $S_n$. The probability \[ Pr_π(G)=Pr(a_{1}a_{2}\cdots a_{n-1}a_{n}=a_{π_{1}}a_{π_{2}}\cdots a_{π_{n-1}}a_{π_{n}}) \] is identical to $P{r}_1^{\omega }(G)$, with \[ ω=a_{1}a_{2}...a_{n-1}a_{n}a_{π_{1}}^{-1}a_{π_{2}}^{-1}\cdots a_{π_{n-1}}^{-1}a_{π_{n}}^{-1}, \] as it is defined in \cite{DasNath1} and \cite{NathDash1}. The notion of commutativity degree, or the probability of a permutation equality $a_1{a}_2=a_2{a}_1$, for which $n=2$ and $\pi =⟨21⟩$, was introduced and assessed by P. Erdös and P. Turan in \cite{ET} in 1968 and by W. H. Gustafson in \cite{G} in 1973. In \cite{G} Gustafson establishes a relation between the probability of $a_1,a_2\in G$ commuting and the number of conjugacy classes in $G$. In this work we define several other parameters, which depend only on a certain interplay between the conjugacy classes of $G$, and compute the probabilities of general permutation equalities in terms of these parameters. It turns out that this probability, for a permutation $\pi$, depends only on the number $c(Gr(\pi ))$ of the alternating cycles in the cycle graph $Gr(\pi )$ of $\pi$. The cycle graph of a permutation was introduced by V. Bafna and P. A. Pevzner in \cite{BP}. We describe the spectrum of the probabilities of permutation equalities in a finite group as $\pi$ varies over all the elements of $S_n$. This spectrum turns-out to be closely related to the partition of $n!$ into a sum of the corresponding Hultman numbers.

36 pages