Intersections of Sylow 2-subgroups in symmetric groups

TL;DR

This paper calculates the asymptotic probability that two random Sylow 2-subgroups in symmetric and alternating groups intersect trivially, providing insights into subgroup interactions in large symmetric groups.

Contribution

It offers the first asymptotic probability computation for trivial intersection of Sylow 2-subgroups in symmetric and alternating groups, complementing recent related research.

Findings

Asymptotic probability of trivial intersection approaches a specific limit

Provides explicit formulas for symmetric and alternating groups

Enhances understanding of subgroup structure in large groups

Abstract

We compute the asymptotic probability that a random pair of Sylow 2-subgroups in $S_n$ or $A_n$ intersects trivially. This calculation complements recent work of Diaconis, Giannelli, Guralnick, Law, Navarro, Sambale, and Spink (see arXiv:2504.01149).