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 labelled 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 Gamma whose vertices are labelled by pairwise distinct primes is the Gruenberg--Kegel graph of only finitely many finite groups, then all such groups are almost simple; moreover, Gamma 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 2G2(27) and E8(2) are uniquely determined by the isomorphism types of their Gruenberg--Kegel graphs.