On the Gruenberg–Kegel graphs of finite groups

Let G be a finite group. The spectrum of G is the set ω(G) of orders of all its elements. The subset of prime elements of ω(G) is called the prime spectrum ofG and is denoted by π(G). The spectrum ω(G) of a groupG defines its Grunberg–Kegel graph (or prime graph) Γ(G) with vertex set π(G), in which any two different vertices r and s are adjacent if and only if the number rs belongs to the set ω(G). We discuss some problems concerning coincidence of Gruenberg–Kegel graphs of non-isomorphic finite groups and of realizability of a graph as the Gruenberg–Kegel graph of a finite group. 1 Terminology and Notation During this paper by “group” we mean “a finite group” and by “graph” we mean “a finite undirected graph without loops and multiple edges”. Our notation and terminology are mostly standard and can be found in [1, 3, 4, 7, 16]. Let G be a group. Denote by π(G) the set of all prime divisors of the order of G and by ω(G) the spectrum of G, i.e. the set of all its element orders. The set ω(G) defines the Gruenberg–Kegel graph (or the prime graph) Γ(G) of G; in this graph the vertex set is π(G) and different vertices p and q are adjacent if and only if pq ∈ ω(G). Denote the number of connected components of Γ(G) by s(G) and the set of connected components of Γ(G) by {πi(G) | 1 ≤ i ≤ s(G)}; for a group G of even order, we assume that 2 ∈ π1(G). Denote by t(G) the greatest cardinality of a coclique in the Gruenberg–Kegel graph of the group G and by t(r,G) the greatest cardinality of a coclique in the Gruenberg–Kegel graph of the group G containing the prime r. Let π be a set of primes. Denote by π′ the set of the primes not in π. Given a natural n, denote by π(n) the set of its prime divisors. Then π(|G|) is exactly π(G) for any group G. If |π(G)| = n then G is called n-primary. A subgroup H of a group G is called a π-Hall subgroup if π(H) ⊆ π and π(|G : H|) ⊆ π′. We will denote by S(G) the solvable radical of a group G (i. e. the largest solvable normal subgroup of G), by F (G) the Fitting subgroup of G (i. e. the largest nilpotent normal subgroup of G), by Soc(G) the socle of G (i. e. the subgroup of G generated by the set of all non-trivial minimal normal subgroups of G) and by Op(G) the largest normal p-subgroup of G. A group G is almost simple if Soc(G) is a simple group. It’s well known, G is almost simple if and only if there exists a simple group S such that S ∼= Inn(S) EG ≤ Aut(S). In this case, S ∼= Soc(G). If G and H are groups, then we use notation G : H (G hH) for a split extension (semidirect product) of G by (with, on) H. Copyright c © by the paper’s authors. Copying permitted for private and academic purposes. In: A.A. Makhnev, S.F. Pravdin (eds.): Proceedings of the 47th International Youth School-conference “Modern Problems in Mathematics and its Applications”, Yekaterinburg, Russia, 02-Feb-2016, published at http://ceur-ws.org