Sylow subgroups for distinct primes and intersection of nilpotent subgroups
Francesca Lisi, Luca Sabatini
TL;DR
This paper investigates how finite groups synchronize the intersections of Sylow subgroups for distinct primes under conjugation, by studying the action of $G$ on the set $\mathcal{DS}(G)$ of Sylow subgroups and relating it to the hypercenter $\mathcal{H}(G)$ and the Fitting subgroup $F(G)$. It proves that a finite group cannot be covered by Sylow normalizers for distinct primes and establishes synchronization results in two-prime cases and in union-like obstructions, providing a foundation for the main conjecture. The authors then prove Conjecture A in two broad settings: symmetric and alternating groups of large degree via probabilistic methods, and metanilpotent groups of odd order via a strengthening of a product-structure argument within $F(G)$. These results connect to longstanding questions about the intersections of nilpotent subgroups and the role of $O_p(G)$ and $F(G)$ in constraining such intersections. Overall, the work advances a unifying framework linking Sylow intersections, nilpotent subgroup behavior, and group actions on Sylow-structures, with concrete bounds and constructive cases that illuminate the conjecture’s validity across families of groups.
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.