Noncyclic Graph of a Group

Abstract

We associate a graph Γ G to a nonlocally cyclic group G (called the noncyclic graph of G) as follows: take G\\ Cyc(G) as vertex set, where Cyc(G) = {x ∈ G| ⟨x, y⟩ is cyclic for all y ∈ G}, and join two vertices if they do not generate a cyclic subgroup. We study the properties of this graph and we establish some graph theoretical properties (such as regularity) of this graph in terms of the group ones. We prove that the clique number of Γ G is finite if and only if Γ G has no infinite clique. We prove that if G is a finite nilpotent group and H is a group with Γ G ≅ Γ H and |Cyc(G)| = |Cyc(H)| = 1, then H is a finite nilpotent group. We give some examples of groups G whose noncyclic graphs are “unique”, i.e., if Γ G ≅ Γ H for some group H, then G ≅ H. In view of these examples, we conjecture that every finite nonabelian simple group has a unique noncyclic graph. Also we give some examples of finite noncyclic groups G with the property that if Γ G ≅ Γ H for some group H, then |G| = |H|. These suggest the question whether the latter property holds for all finite noncyclic groups.