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A finite group G is said to have the m-CI property if, for any two Cayley graphs Cay(G, S) and Cay(G, T) of valency m, Cay(G, S) ≅ Cay(G, T) implies Sσ = T for some automorphism σ of G. In this paper, we investigate finite groups with the m-CI property. We first construct groups with the 3-CI property but not with the 2-CI property, and then prove that a nonabelian simple group has the 3-CI property if and only if it is A5. Finally, for infinitely many values of m, we construct Frobenius groups with the m-CI property but not with the nontrivial k-CI property for any k < m.

The solution to the more basic problem of classifying all finite simple groups was completed in 1980 and it is now the job of mathematicians in related fields to determine the consequences of this classification on their field of research. The purpose of this paper is to survey several fairly basic problems in the theory of finite permutation groups where the simple group classification has led to a satisfactory solution.

Let p denote a prime, G a finite p′-group and A an elementary atelian group of operators on G. Suppose that A has order p3 and that if w ∈ A# then CG(w) is nilpotent. It is proved that G is nilpotent.