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Further restrictions on the structure of finite CI-groups. (English) Zbl 1124.05045
Summary: A group $$G$$ is called a CI-group if, for any subsets $$S,T\subset G$$, whenever two Cayley graphs Cay$$(G,S)$$ and Cay$$(G,T)$$ are isomorphic, there exists an element $$\sigma \in \text{Aut}(G)$$ such that $$S\sigma = T$$. The problem of seeking finite CI-groups is a long-standing open problem in the area of Cayley graphs. This paper contributes towards a complete classification of finite CI-groups. First it is shown that the Frobenius groups of order $$4p$$ and $$6p$$, and the metacyclic groups of order $$9p$$ of which the centre has order 3 are not CI-groups, where $$p$$ is an odd prime. Then a shorter explicit list is given of candidates for finite CI-groups. Finally, some new families of finite CI-groups are found, that is, the metacyclic groups of order $$4p$$ (with centre of order 2) and of order $$8p$$ (with centre of order 4) are CI-groups, and a proof is given for the Frobenius group of order $$3p$$ to be a CI-group, where $$p$$ is a prime.
Reviewer: Reviewer (Berlin)

MSC:
 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
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