Sylow subgroups for distinct primes and intersection of nilpotent subgroups

TL;DR

This paper investigates the intersections of Sylow subgroups in finite groups, proposing a conjecture about their minimal intersections and proving it in specific classes like symmetric, alternating, and metanilpotent groups.

Contribution

It introduces a conjecture on minimal intersections of Sylow subgroups and proves it for symmetric, alternating, and metanilpotent groups, advancing understanding of subgroup intersections.

Findings

Finite groups cannot be covered by Sylow normalizers for distinct primes.

The conjecture holds for symmetric and alternating groups of large degree.

The conjecture is also valid for metanilpotent groups of odd order.

Abstract

Let $G$ be a finite group and let $(P_i)_{i=1}^n$ be Sylow subgroups for distinct primes $p_1,\ldots,p_n$. We conjecture that there exists $x \in G$ such that $P_i \cap P_i^x$ is inclusion-minimal in $\{ P_i \cap P_i^g : g \in G\}$ for all $i$. As a first step in this direction, we show that a finite group cannot be covered by (proper) Sylow normalizers for distinct primes. Then we settle the conjecture in two opposite situations: symmetric and alternating groups of large degree and metanilpotent groups of odd order. Applications concerning the intersections of nilpotent subgroups are discussed.