Realizing Gruenberg-Kegel graphs of $T$-solvable groups with structurally simplified extensions of $T$

TL;DR

This paper studies the prime graphs of T-solvable groups, proving realizability of their complements for many simple groups and classifying specific cases like PSL(2,13)-solvable groups.

Contribution

It demonstrates that prime graph complements of T-solvable groups are realizable by certain structured groups for a large class of simple groups and provides a classification for PSL(2,13)-solvable groups.

Findings

Prime graph complements are realizable for many non-abelian simple groups.

Classification of prime graph complements for PSL(2,13)-solvable groups.

Conjecture on the general realizability of prime graph complements.

Abstract

Given a finite group $G$, its prime graph $\Gamma(G)$ (also known as its Gruenberg-Kegel graph) is the graph whose vertices are the prime divisors of $|G|$ and where edges $\{p, q\}$ exist whenever $G$ contains an element of order $pq$. We continue the study of prime graphs for $T$-solvable groups; that is, groups whose composition factors are either abelian or isomorphic to some fixed non-abelian simple group $T$. For a large class of non-abelian simple groups $T$, we prove that the prime graph complements of $T$-solvable groups are always realizable by a solvable group and a quasi simple or almost simple $T$-solvable group acting by automorphisms on a direct product of elementary abelian groups. We conjecture that a similar result holds in full generality. Moreover, we apply our result to classify in purely graph-theoretic terms the prime graph complements of $\operatorname{PSL}(2,13)$-solvable groups, and indicate other interesting classes of groups matching the assumptions of our main theorem.