Direct sum of CI-groups

Gábor Somlai
Ben Gurion University of Negev

Misha Muzychuk
Netanya Academic College

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Minisymposium: APPLICATIONS OF GROUPS IN GRAPH THEORY

Content: The investigation of CI-groups started with the conjecture of \'Ad\'am. Using the terminology later introduced by Babai, he conjectured that every cyclic group is a CI-group. A group $G$ is called a CI-group when two Cayley graphs of $G$ are isomorphic only if there is an automorphism of the group $G$ which induces an isomorphism between the two Cayley graphs. It was proved by Babai and Frankl that a CI-group, which is also a $p$-group can only be elementary abelian $p$-group, quaternion group or cyclic group of small order. Several authors contributed to the description of CI-groups and a fairly short list of possible CI-groups was described by Li, Lu and P\'alfy. Most of the candidates are abelian groups. Kov\'acs and Muzychuk conjectured that the direct sum of CI-groups of relative prime order is a CI-group. The list provided by Li, Lu and P\'alfy shows that this is one of the main questions in this area. We contribute towards this conjecture by proving that if $H$ is an abelian CI-group, $q$ is a prime which does not divide $|H|$, then $H \times \mathbb{Z}_q$ is a CI-group. The proof relies on two different approaches. One of them uses elementary permutation theory tools worked out in a previous paper about CI-groups of order $p^3q$ but the most important technique is the theory of Schur rings used by Kov\'acs and Muzychuk. \\ Joint work with Mikhail Muzychuk.

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