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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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