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

TL;DR

This work advances the understanding of Gruenberg-Kegel graphs by proving that the prime graph complements of finite $T$-solvable groups with certain structural constraints can be realized by solvable groups acting on a direct product of elementary abelian groups, together with a $T$-solvable quasi-simple or almost simple component. The authors reduce the problem to modular representations of these components and extend the framework to outer automorphisms via a 2-group assumption on Out$(T)$, enabling a unified realization $\overline{\Gamma}(G)=\overline{\Gamma}(A\rtimes(H\times K))$ or $\overline{\Gamma}(A\rtimes(E\times K))$. They develop a robust set of tools—general lemmas on Frobenius-type structures, a reduction for $G\cong N.T$, and an induction to outer automorphisms—to realize a wide class of prime graph complements and apply the main theorem to classify $PSL(2,13)$-solvable groups. The results yield both broad applicability to several families of groups (e.g., $A_{11},\dots,G_2(4)$) and a concrete, complete classification for PSL$(2,13)$-solvable groups, illustrating the structural approach’s effectiveness and suggesting a general conjecture that all such graphs admit realizations of this restricted form. These insights provide a pathway toward systematic, graph-theoretic classifications of finite groups via modular representation theory and controlled extensions.

Abstract

Given a finite group $G$, its prime graph $Γ(G)$ (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.

Figures (2)

• Figure 1: An example of $\vec{\Gamma}$ as given in \ref{['monochromaticneighbors']}
• Figure 2: A graph satisfying Condition 2(b)(i) of \ref{['pslclassification']} with triangles highlighted.

Theorems & Definitions (97)

• Theorem 1: 2015REU
• Theorem \ref{pslclassification}
• Theorem A
• Conjecture \ref{pslclassification}
• Theorem \ref{thmapplicablegroups}
• Corollary \ref{thmapplicablegroups}
• Definition \ref{thmapplicablegroups}
• Definition \ref{thmapplicablegroups}
• Definition \ref{thmapplicablegroups}
• Lemma \ref{thmapplicablegroups}