Symmetric functions, generalized blocks, and permutations with restricted cycle structure (Q872055)

Let \(S_n\) be the symmetric group of degree \(n\in \mathbb{N}\), and let \(\ell\in \mathbb{N}\), \(\ell\geq 2\). An element \(g\in S_n\) is called \(\ell\)-regular if no cycle of \(g\) has a length which is divisible by \(\ell\). Recently \textit{B. Külshammer, J. B. Olsson} and \textit{G. R. Robinson} [Invent. Math. 151, 513-552 (2003; Zbl 1043.20007)] have used this notion in order to develop a theory of \(\ell\)-blocks of \(S_n\). Motivated by their paper, the author studies the proportion of \(\ell\)-regular elements in \(S_n\), reproving a classical result of Erdös and Turán. He also shows that, in an arbitrary permutation group \(G\) of degree \(n\), the proportion of \(\ell\)-regular elements has the form \(\frac{\alpha_\ell(G)}{\ell^w \cdot(w!)_\pi}\) where \(\alpha_\ell(G)\) is an integer, \(w=\left\lfloor \frac{n}{\ell}\right\rfloor\) and \(\pi\) is the set of prime divisors of \(\ell\). This leads to a formula (originally proved by Beals, Leedham-Green, Niemeyer, Praeger and Seress), for the proportion of \(\ell\)-regular elements in the alternating group \(A_n\). The author also obtains a formula for the average number of fixed points of an \(\ell\)-regular element in \(S_n\), and a number of related results.