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

• Published in: Communications in mathematics and statistics Vol. 10; no. 4; pp. 653 - 667
• Main Authors: Kondrat’ev, A. S., Minigulov, N. A.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2022; Springer Nature B.V
• Subjects: Apexes; Graph theory; Graphs; Group theory; Mathematics; Mathematics and Statistics; Statistics; Vertex sets; Non-solvable group; Finite group; The paw; Gruenberg–Kegel graph; 20D60; 20D06; 05C25
• ISSN: 2194-6701; 2194-671X
• DOI: 10.1007/s40304-021-00242-x

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 of Γ ( G ) divides the order of the solvable radical of G .