Classification of the Prime Graphs of $\operatorname{Sz}(8)$-, $\operatorname{Sz}(32)$-, and $\operatorname{PSL}(2, 2^5)$-Solvable Groups

TL;DR

This paper completes the classification of prime graphs for certain solvable groups with specific simple composition factors, including the challenging cases of Sz(8), Sz(32), and PSL(2, 2^5), involving new techniques like Brauer character tables.

Contribution

It provides the first classification results for groups where the outer automorphism group has prime factors not dividing the group order, advancing understanding of prime graphs in complex cases.

Findings

Classified prime graphs for Sz(8), Sz(32), and PSL(2, 2^5) groups.

Introduced methods involving Brauer character tables for these classifications.

Resolved previously open cases in the classification of prime graphs of solvable groups.

Abstract

For a finite group $G$, the vertices of the prime graph $\Gamma(G)$ are the primes that divide $|G|$, and two vertices $p$ and $q$ are connected by an edge if there is an element of order $pq$ in $G$. Prime graphs of solvable groups have been classified, and prime graphs of groups whose noncyclic composition factors are isomorphic to a single nonabelian simple group $T$ have been classified in the case where $T$ has order divisible by exactly three or four distinct primes, except for the cases $T = \operatorname{Sz}(8)$, $T = \operatorname{Sz}(32)$, and $T = \operatorname{PSL}(2,q)$, which in some sense are the hardest cases. In this paper, we complete the classification for $T = \operatorname{Sz}(32)$, $T = \operatorname{Sz}(8)$, and $T = \operatorname{PSL}(2,2^5)$, with the latter two being the first cases ever studied where $|\text{Out}(T)|$ has prime factors which do not divide $|T|$. The groups studied in this paper are also the first ones requiring knowledge of their Brauer character tables to complete the classification task.