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Intersections of Sylow 2-subgroups in symmetric groups

Sean Eberhard

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).

Intersections of Sylow 2-subgroups in symmetric groups

Abstract

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

Paper Structure

This paper contains 3 theorems, 17 equations.

Key Result

Theorem 1

Let $P \in \operatorname{Syl}_2(S_n)$ and let $x \in S_n$ be chosen uniformly at random. Then If $P \in \operatorname{Syl}_2(A_n)$ then similarly

Theorems & Definitions (7)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • proof : Proof of \ref{['thm']}
  • Remark 4