Non-cyclic graph associated with a group

Abstract

We associate a graph CG to a non locally cyclic group G (called the non-cyclic 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} is called the cyclicizer of G, and join two vertices if they do not generate a cyclic subgroup. For a simple graph Γ, w(Γ) denotes the clique number of Γ, which is the maximum size (if it exists) of a complete subgraph of Γ. In this paper we characterize groups whose non-cyclic graphs have clique numbers at most 4. We prove that a non-cyclic group G is solvable whenever w(CG) < 31 and the equality for a non-solvable group G holds if and only if G/Cyc(G) ≅ A5 or S5. © World Scientific Publishing Company.