Non-cyclic graph associated with a group

TL;DR

This paper studies the non-cyclic graph associated with a group, characterizing groups with small clique numbers and establishing conditions under which the group is solvable or related to specific simple groups.

Contribution

It introduces the non-cyclic graph of a group and characterizes groups with clique numbers at most 4, also linking clique number bounds to solvability and specific simple groups.

Findings

Groups with non-cyclic graph clique number ≤ 4 are characterized.

A non-cyclic group is solvable if the clique number of its non-cyclic graph is less than 31.

Non-solvable groups with maximum clique number 31 are isomorphic to A_5 or S_5 modulo the cyclicizer.

Abstract

We associate a graph $\mathcal{C}_G$ to a non locally cyclic group $G$ (called the non-cyclic graph of $G$) as follows: take $G\backslash Cyc(G)$ as vertex set, where $Cyc(G)=\{x\in G | < x,y> \text{is cyclic for all} y\in G\}$ is called the cyclicizer of $G$, and join two vertices if they do not generate a cyclic subgroup. For a simple graph $\Gamma$, $w(\Gamma)$ denotes the clique number of $\Gamma$, which is the maximum size (if it exists) of a complete subgraph of $\Gamma$. 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 $w(\mathcal{C}_G)<31$ and the equality for a non-solvable group $G$ holds if and only if $G/Cyc(G)\cong A_5$ or $S_5$.