Criterion of unrecognizability of a finite group by its Gruenberg–Kegel graph

Abstract

The Gruenberg–Kegel graph Γ(G) associated with a finite group G is an undirected graph without loops and multiple edges whose vertices are the prime divisors of |G| and in which vertices p and q are adjacent in Γ(G) if and only if G contains an element of order pq. This graph has been the subject of much recent interest; one of our goals here is to give a survey of some of this material, relating to groups with the same Gruenberg–Kegel graph. However, our main aim is to prove several new results. Among them are the following. • There are infinitely many finite groups with the same Gruenberg–Kegel graph as the Gruenberg–Kegel of a finite group G if and only if there is a finite group H with non-trivial solvable radical such that Γ(G)=Γ(H). • There is a function F on the natural numbers with the property that if a finite n-vertex graph whose vertices are labeled by pairwise distinct primes is the Gruenberg–Kegel graph of more than F(n) finite groups, then it is the Gruenberg–Kegel graph of infinitely many finite groups. (The function we give satisfies F(n)=O(n7), but this is not best possible.) • If a finite graph Γ whose vertices are labeled by pairwise distinct primes is the Gruenberg–Kegel graph of only finitely many finite groups, then all such groups are almost simple; moreover, Γ has at least three pairwise non-adjacent vertices, and each vertex is non-adjacent to at least one other vertex, in particular, 2 is non-adjacent to at least one odd vertex. • Groups whose power graphs, or commuting graphs, are isomorphic have the same Gruenberg–Kegel graph. • The groups G22(27) and E8(2) are uniquely determined by the isomorphism types of their Gruenberg–Kegel graphs. In addition, we consider groups whose Gruenberg–Kegel graph has no edges. These are the groups in which every element has prime power order, and have been studied under the name EPPO groups; completing this line of research, we give a complete list of such groups. © 2021 Elsevier Inc.