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