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

The Gruenberg-Kegel graph(or the prime graph)Γ(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 ofΓ(G)divides the order of the solvable radical of G.