Classifying prime graphs of finite groups -- a methodical approach

Abstract

For a finite group $G$, the vertices of the prime graph $Γ(G)$ are the primes that divide $|G|$, and two vertices $p$ and $q$ are connected by an edge if and only if there is an element of order $pq$ in $G$. Prime graphs of solvable groups as well as groups whose noncyclic composition factors have order divisible by exactly three distinct primes have been classified in graph-theoretic terms. In this paper, we begin to develop a general theory on the existence of edges in the prime graph of an arbitrary $T$-solvable group, that is, a group whose composition factors are cyclic or isomorphic to a fixed nonabelian simple group $T$. We then apply these results to classify the prime graphs of $T$-solvable groups for, in a suitable sense, most $T$ such that $|T|$ has exactly four prime divisors. We find that these groups almost always have a 3-colorable prime graph complement containing few possible triangles.