On counting centralizer subgroups of symmetric groups

TL;DR

This paper investigates the structure of intersections of conjugates of a specific centralizer subgroup in symmetric groups, showing that permutations producing polynomial-sized intersections are extremely rare.

Contribution

It characterizes the distribution of intersection sizes of conjugate centralizer subgroups in symmetric groups, revealing that polynomial-sized intersections are of negligible density.

Findings

Permutations with polynomial-sized intersections have density zero.

The structure of $gHg^{-1}igcap H$ is analyzed for all $g$ in $S_{2m}$.

Most conjugates produce intersections of sublinear size.

Abstract

Let $S_{2m}$ be the symmetric group, $h=(1\ 2)(3\ 4)\cdots(2m-1\ 2m)$ and $H=C(h)$. We consider the structure of $gHg^{-1}\cap H$ for any $g\in S_{2m}$. We prove the permutations $g$ which makes $gHg^{-1}\cap H$ have size of polynomial in $m$ have density zero.