Finite Almost Simple $4$-Primary Groups with Connected Gruenberg–Kegel Graph

Abstract

Let $G$ be a finite group. Denote by $\pi(G)$ the set of prime divisors of the order of $G$. The Gruenberg–Kegel graph (prime graph) of $G$ is the graph with the vertex set $\pi(G)$ in which two different vertices $p$ and $q$ are adjacent if and only if $G$ has an element of order $pq$. If $|\pi(G)|=n$, then the group $G$ is called $n$-primary. In 2011, A.S. Kondrat’ev and I.V. Khramtsov described finite almost simple $4$-primary groups with disconnected Gruenberg–Kegel graph. In the present paper, we describe finite almost simple $4$-primary groups with connected Gruenberg–Kegel graph. For each of these groups, its Gruenberg–Kegel graph is found. The results are presented in a table . According to the table, there are $32$ such groups. The results are obtained with the use of the computer system GAP.