# A finite simple group is CCA if and only if it has no element of order four

## TL;DR

This paper characterizes finite simple groups as CCA if and only if they lack elements of order four, and demonstrates that many 2-groups are non-CCA, advancing understanding of Cayley graph automorphisms.

## Contribution

It provides a complete characterization of finite simple groups that are CCA based on element orders, and shows that many 2-groups are non-CCA.

## Key findings

- Finite simple groups are CCA iff they have no element of order four.
- Many 2-groups are non-CCA.
- The characterization links group element orders to Cayley graph automorphism properties.

## Abstract

A Cayley graph for a group $G$ is CCA if every automorphism of the graph that preserves the edge-orbits under the regular representation of $G$ is an element of the normaliser of $G$. A group $G$ is then said to be CCA if every connected Cayley graph on $G$ is CCA. We show that a finite simple group is CCA if and only if it has no element of order 4. We also show that "many" 2-groups are non-CCA.