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Cayley integral simple group

A group G {\displaystyle G} is Cayley integral simple (CIS) if the connected Cayley graph Γ (G, S) {\displaystyle \Gamma (G,S)} is integral exactly when the symmetric generating set S {\displaystyle S} is the complement of a subgroup of G {\displaystyle G}. A result of Ahmady, Bell, and Mohar shows that all CIS groups are isomorphic to Z / p Z, Z / p 2 Z {\displaystyle \mathbb {Z} /p\mathbb {Z},\mathbb {Z} /p^{2}\mathbb {Z} }, or Z 2 × Z 2 {\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{2}} for primes p {\displaystyle p}. It is important that S {\displaystyle S} actually generates the entire group G {\displaystyle G} in order for the Cayley graph to be connected. (If S {\displaystyle S} does not generate G {\displaystyle G}, the Cayley graph may still be integral, but the complement of S {\displaystyle S} is not necessarily a subgroup.)

In the example of G = Z / 5 Z {\displaystyle G=\mathbb {Z} /5\mathbb {Z} }, the symmetric generating sets (up to graph isomorphism) are

The only subgroups of Z / 5 Z {\displaystyle \mathbb {Z} /5\mathbb {Z} } are the whole group and the trivial group, and the only symmetric generating set S {\displaystyle S} that produces an integral graph is the complement of the trivial group. Therefore Z / 5 Z {\displaystyle \mathbb {Z} /5\mathbb {Z} } must be a CIS group.

The proof of the complete CIS classification uses the fact that every subgroup and homomorphic image of a CIS group is also a CIS group.