Criteria for Classifying Prime Graphs of PSL(2, q)-Solvable Groups

For a finite group $G$, the prime graph $\mathrm{\Gamma }(G)$ (also known as Gruenberg-Kegel graph) is defined to be the graph where the vertices are the primes that divide $\mathrm{\mid }G\mathrm{\mid }$ such that two vertices $p$ and $q$ share an edge if and only if there is an element of order $pq$ in $G$. The prime graphs of solvable groups have been classified. The prime graphs of groups whose noncyclic composition factors are isomorphic to a single nonabelian simple group $T$ where $\mathrm{\mid }T\mathrm{\mid }$ is divisible by three or four distinct primes have been classified except for the cases where $T=\mathrm{PSL}(2,q)$ for $q\mathrm{\ne }{2}^5$ and $\mathrm{\mid }\mathrm{PSL}(2,q)\mathrm{\mid }$ is divisible by exactly four primes. In this paper, we provide criteria for general classification results for certain classes of $T$, and then use them to classify the prime graphs of some $T$-solvable groups for $T$ a suitably small $\mathrm{PSL}(2,q)$-group. We also provide general results on the prime graphs of $T$-solvable groups where $T$ is a member of the possibly infinite family of groups $\mathrm{PSL}(2,2^f)$ such that $f\ge 5,f$ is prime, and $\mathrm{\mid }\mathrm{PSL}(2,2^f)\mathrm{\mid }$ is divisible by exactly four primes. This is the first paper to prove general results about the prime graphs of $T$-solvable groups where $T$ belongs to a large (probably infinite) family of groups.