On Finite Non-Solvable Groups Whose Gruenberg-Kegel Graphs are Isomorphic to the Paw

The Gruenberg-Kegel graph (or the prime graph) G (G) of a finite group G is a graph, in which 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. The paw is a graph on four vertices whose degrees are 1, 2, 2, 3. We consider the problem of describing finite groups whose Gruenberg-Kegel graphs are isomorphic as abstract graphs to the paw. For example, the Gruenberg-Kegel graph of the alternating group A(10) of degree 10 is isomorphic as abstract graph to the paw. In this paper, we describe finite non-solvable groups G whose Gruenberg-Kegel graphs are isomorphic as abstract graphs to the paw in the case when G has no elements of order 6 or the vertex of degree 1 of G (G) divides the order of the solvable radical of G.