Editing Cayley graph (section)

=== Cayley integral group ===

A slightly different notion is that of a Cayley integral group <math>G</math>, in which every symmetric subset <math>S</math> produces an integral graph <math>\Gamma(G,S)</math>. Note that <math>S</math> no longer has to generate the entire group.

The complete list of Cayley integral groups is given by <math>\mathbb{Z}_2^n\times \mathbb{Z}_3^m,\mathbb{Z}_2^n\times \mathbb{Z}_4^n, Q_8\times \mathbb{Z}_2^n,S_3</math>, and the dicyclic group of order <math>12</math>, where <math>m,n\in \mathbb{Z}_{\ge 0}</math> and <math>Q_8</math> is the quaternion group.<ref name="CIS"/> The proof relies on two important properties of Cayley integral groups:
* Subgroups and homomorphic images of Cayley integral groups are also Cayley integral groups.
* A group is Cayley integral iff every connected Cayley graph of the group is also integral.