On the Sylow graph of a group and Sylow normalizers

TL;DR

This paper introduces the Sylow graph of a finite group, studies its properties for almost simple groups, and demonstrates how it can be used to infer structural information from Sylow normalizers.

Contribution

It defines the Sylow graph and proves its connectivity and diameter bound for almost simple groups, linking Sylow normalizers to group structure analysis.

Findings

The Sylow graph is connected for almost simple groups.

The diameter of the Sylow graph is at most 5.

Applications to group structure inference from Sylow normalizers.

Abstract

Let $G$ be a finite group and $G_p$ be a Sylow $p$-subgroup of $G$ for a prime $p$ in $\pi(G)$, the set of all prime divisors of the order of $G$. The automiser $A_p(G)$ is defined to be the group $N_G(G_p)/G_pC_G(G_p)$. We define the Sylow graph $\Gamma_A(G)$ of the group $G$, with set of vertices $\pi(G)$, as follows: Two vertices $p,q\in\pi(G)$ form an edge of $\Gamma_A(G)$ if either $q\in\pi(A_p(G))$ or $p\in \pi(A_q(G))$. The following result is obtained: Theorem: Let $G$ be a finite almost simple group. Then the graph $\Gamma_A(G)$ is connected and has diameter at most 5. We also show how this result can be applied to derive information on the structure of a group from the normalizers of its Sylow subgroups.