Congruences between word length statistics for the finitary alternating and symmetric groups

TL;DR

This paper explores congruence relationships between the conjugacy growth series of finitary symmetric and alternating groups, revealing systematic congruences modulo powers of primes like 5 and 7, extending to all primes ≥ 5.

Contribution

It establishes new congruence relations between the growth series of these groups using advanced partition function congruences, generalizing previous results to all primes ≥ 5.

Findings

Existence of congruences modulo powers of 5 and 7 between the series.

Systematic description of these congruences.

Extension of congruence relations to all primes ≥ 5.

Abstract

In a recent paper, Bacher and de la Harpe study conjugacy growth series of infinite permutation groups and their relationships with $p(n)$, the partition function, and $p(n)_{\textbf{e}}$, a generalized partition function. They prove identities for the conjugacy growth series of the finitary symmetric group and the finitary alternating group. The group theory also motivates an investigation into congruence relationships between the finitary symmetric group and the finitary alternating group. Using the Ramanujan congruences for the partition function $p(n)$ and Atkin's generalization to the $k$-colored partition function $p_{k}(n)$, we prove the existence of congruence relations between these two series modulo arbitrary powers of 5 and 7, which we systematically describe. Furthermore, we prove that such relationships exist modulo powers of all primes $\ell\geq 5$.