Prime Graphs of Some K_4-Solvable Groups

We define the prime graph $\Gamma (G)$ of a group $G$ as follows: The vertices are the prime divisors of $|G|$, and two vertices $p$ and $q$ are connected by an edge if and only if there exists an element of order $pq$ in $G$. A complete classification of prime graphs of finite groups remains elusive, yet progress has been made for some families of groups. For example, it has been shown that a graph is the prime graph of a solvable group if and only if its complement is triangle-free and 3-colorable. We hope to find similar classifications for groups with nonabelian composition factors. To that end, we make two definitions. First, given a nonabelian simple group $T$, we say that a group is $T$-solvable if its composition factors are either cyclic or isomorphic to $T$. Second, we say that a simple group is $K_n$ if its order has exactly $n$ prime divisors. The prime graphs of groups with $K_1$ or $K_2$ composition factors are well-understood because such groups are always solvable. Last year, a team of undergraduate researchers classified the prime graphs of $T$-solvable groups for all $K_3$ groups $T$. Now, we present further classification results for $T$-solvable groups, this time selecting $T$ from a large family of $K_4$ groups.