Finite non-solvable groups whose Gruenberg–Kegel graphs are isomorphic to the paw. Case $q\leq 3$

A. S. Kondrat'ev^a, N. A. Minigulov^ab, M. S. Nirova<sup>c

a</sup> N. N. Krasovskii Institute of Mathematics and Mechanics UB RAS, Ural Mathematical Center, 16 S. Kovalevskaya St., Yekaterinburg 620108, Russia
^b Ural Federal University, 19 Mira St., Ekaterinburg 620062, Russia
^c Kabardino-Balkarian State University, 173 Chernyshevsky St., Nalchik 360004, Russia

Abstract: The Gruenberg–Kegel graph $\Gamma(G)$ (or the prime graph) of a finite group $G$ is the graph such that the vertex set is the set of all prime divisors of the order of $G$ and two different vertices $p$ and $q$ are adjacent if and only if there exists an element of order $pq$ in $G$. One of popular directions of researches in finite group theory is the study of finite groups with given properties of their Gruenberg–Kegel graphs. In 2012–2013 the first author described finite groups with the Gruenberg–Kegel graph as for the group ${\rm Aut}(J_2)$ and as for the group $A_{10}$. The Gruenberg–Kegel graphs of groups $A_{10}$ and ${\rm Aut}(J_2)$ are isomorphic (as abstract graphs) to the paw. The paw is the graph with four vertices whose degrees are $1$, $2$, $2$, and $3$. Generalizing these results, we consider the problem of describing finite groups such that the Gruenberg–Kegel graphs of these groups are isomorphic to the paw. In 2018 Kondrat'ev and Minigulov proved that if $G$ is a finite non-solvable group and the graph $\Gamma(G)$ is isomorphic to the paw, then the quotion group $G/S(G)$ of the group $G$ by its solvable radical $S(G)$ is almost simple. Moreover, they classified all finite almost simple groups $G$ such that the graphs $\Gamma(G)$ of these groups are isomorphic to subgraphs of the paw. In 2022 Kondrat'ev and Minigulov described all finite solvable groups such that the Gruenberg–Kegel graph is isomorphic to the paw. Moreover, they classified finite non-solvable groups $G$, where the Gruenberg–Kegel graphs of these groups are isomorphic to the paw, in the following cases: $(1)$ $G$ does not contain elements of order $6$; $(2)$ $G$ has an element of order $6$ and the vertex of degree $1$ of the graph $\Gamma(G)$ divides $|S(G)|$. In this manuscript we continue the investigation of this problem and study its important new case of a finite non-solvable group $G$ such that the Gruenberg–Kegel graph of this group is isomorphic to the paw, where the vertex of degree $1$ of the graph $\Gamma(G)$ does not exceed $3$.

Key words: finite group, non-solvable group, Gruenberg–Kegel graph, paw.

UDC: 512.542

MSC: 20D10, 20D60, 05C25

Received: 27.04.2025

DOI: 10.46698/o5301-6902-4904-l