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Let G be a finite group and S a Cayley subset of G (that is, 1G ∉ S). If, for all T ⊂ G, Cay Cay(G,T) implies that Sα = T for some
, then S is called a CI-subset of G. For a group G and a positive integer m, if all Cayley subsets of size m are CI-subsets, then G is said to have the m-DCI property. In this paper, all finite groups which have the 2-DCI property but do not have the 1-;DCI property are completely classified
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