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.