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

TL;DR

This work develops a systematic framework for classifying prime graphs and their complements of $T$-solvable groups with $T$ a PSL$(2,q)$-type simple group. It introduces a general realizability criterion for rooted four-vertex graphs via Brauer-fixed-point data and the Frobenius digraph, then applies it to several PSL families, including PSL$(2,2^4)$, PSL$(2,3^3)$, PSL$(2,7^2)$, PSL$(2,11)$, PSL$(2,19)$, and PSL$(2,23)$, to obtain complete classifications of their solvable prime graphs. The paper also treats PSL$(2,2^f)$ with $f\ge5$ to derive edge-structure restrictions and partial realizability results for graphs on four and five vertices, supported by modular representation data and Suzuki-type arguments. Collectively, these results reduce the problem of classifying prime graphs of a broad, potentially infinite family of PSL$(2,q)$-solvable groups to a combination of structural group theory and computational checks on Sylow subgroups and Brauer character data, with several open cases and a conjecture guiding future work.

Abstract

For a finite group $G$, the prime graph $Γ(G)$ (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.

Figures (38)

• Figure : $\mathop{\mathrm{\overline{\Gamma}}}\nolimits(T)$
• Figure : $\mathop{\mathrm{\overline{\Gamma}}}\nolimits\left(\mathop{\mathrm{PSL}}\nolimits(2,2^4)\right)$
• Figure : $\mathop{\mathrm{\overline{\Gamma}}}\nolimits(T)$
• Figure : $\mathop{\mathrm{\overline{\Gamma}}}\nolimits(\mathop{\mathrm{PSL}}\nolimits(2,3^3))$
• Figure : $\mathop{\mathrm{\overline{\Gamma}}}\nolimits(\mathop{\mathrm{PSL}}\nolimits(2,7^2))$

Theorems & Definitions (125)

• Definition 1
• Definition 2
• Definition 3
• Theorem 4
• Definition 5
• Definition 6
• Definition 7
• Definition 8
• Definition 9
• Lemma 10