A.S. Kondrat’ev. On the recognizability of sporadic simple groups $Ru$, $HN$, $Fi_{22}$, $He$, $M^cL$, and $Co_3$ by the Gruenberg–Kegel graph ... P. 79-87

The Gruenberg-Kegel graph (prime graph) $\Gamma(G)$ of a finite group $G$ is a graph in which the vertices are the prime divisors of the order of $G$ and two distinct vertices $p$ and $q$ are adjacent if and only if $G$ contains an element of order $pq$. The problem of recognition of finite groups by the Gruenberg-Kegel graph is of great interest in the finite group theory. For a finite group $G$, $h_{\Gamma}(G)$ denotes the number of all pairwise nonisomorhic finite groups $H$ such that $\Gamma(H)=\Gamma(G)$ (if the set of such groups $H$ is infinite, we write $h_{\Gamma}(G)=\infty$). A group $G$ is called $n$-recognizable by the Gruenberg-Kegel graph if $h_{\Gamma}(G)=n<\infty$, recognizable by the Gruenberg-Kegel graph if $h_{\Gamma}(G)=1$, and unrecognizable by the Gruenberg-Kegel graph if $h_{\Gamma}(G)=\infty$. We say that the problem of recognizability by the Gruenberg-Kegel graph is solved for a finite group $G$ if the value $h_{\Gamma}(G)$ is found. For a finite group $G$ unrecognizable by the Gruenberg- Kegel graph, the question of the (normal) structure of finite groups with the same Gruenberg- Kegel graph as $G$ is also of interest. In 2003, M. Hagie investigated the structure of finite groups having the same Gruenberg-Kegel graph as some sporadic simple group. In particular, she gave first examples of finite groups recognizable by the Gruenberg-Kegel graph; they were the sporadic simple groups $J_1$, $M_{22}$, $M_{23}$, $M_{24}$, and $Co_2$. However, that investigation was not completed. In 2006, A. V. Zavarnitsine established that the group $J_4$ is recognizable by the Gruenberg-Kegel graph. The unrecognizability of the sporadic groups $M_{12}$ and $J_2$ was known previously; it follows from the unrecognizability of these groups by the spectrum. In the present paper, we continue Hagie's study and use her results. For any sporadic simple group $S$ isomorphic to $Ru$, $HN$, $Fi_{22}$, $He$, $M^cL$, or $Co_3$, we find all the finite groups having the same Gruenberg-Kegel graph as $S$. Thus, for these six groups, we complete Hagie's investigation and, in particular, we solve the problem of recognizability by the Gruenberg-Kegel graph.

Keywords: finite group, simple group, sporadic group, spectrum, Gruenberg-Kegel graph, recognition by the Gruenberg-Kegel graph

Received September 9, 2019

Revised November 19, 2019

Accepted November 21, 2019

Anatolii Semenovich Kondrat’ev, Dr. Phys.-Math. Sci., N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia, e-mail: A.S.Kondratiev@imm.uran.ru

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Cite this article as: A.S.Kondrat’ev. On the recognizability of sporadic simple groups $Ru$, $HN$, $Fi_{22}$, $He$, $M^cL$, and $Co_3$ by their Gruenberg–Kegel graphs, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 4, pp. 79–87; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2021, Vol. 313, Suppl. 1, pp. S125–S132.