Non-cyclic graph associated with a group

Details Non-cyclic graph associated with a group Non-cyclic graph associated with a group

2008-10-02

arxiv.org/abs/0810.0345v1

Mathematics

Group Theory

to appear in Journal of Algebra and its Applications

Scientific paper

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 $S_5$.

Affiliated with

Also associated with

No associations

Rating

Non-cyclic graph associated with a group does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Non-cyclic graph associated with a group, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Non-cyclic graph associated with a group will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-491891