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

TL;DR

This paper develops criteria to classify prime graphs of certain PSL(2, q)-solvable groups, extending understanding of their structure and providing the first general results for an infinite family of such groups.

Contribution

It introduces new classification criteria for prime graphs of PSL(2, q)-solvable groups, advancing the understanding of their structure and properties.

Findings

Classified prime graphs for some PSL(2, q)-solvable groups.

Established general results for prime graphs where T is PSL(2, 2^f) with f prime.

First to prove broad results for prime graphs of an infinite family of T-sovable groups.

Abstract

For a finite group $G$, the prime graph $\Gamma(G)$ (also known as Gruenberg-Kegel graph) is defined to be the graph where the vertices are the primes that divide $|G|$ 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 $|T|$ is divisible by three or four distinct primes have been classified except for the cases where $T = \operatorname{PSL}(2,q)$ for $q\neq 2^5$ and $|\operatorname{PSL}(2,q)|$ 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 $\operatorname{PSL}(2, q)$-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 $\operatorname{PSL}(2, 2^f)$ such that $f\geq 5, f$ is prime, and $|\operatorname{PSL}(2, 2^f)|$ 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.