We consider the distance enumerator ∆G(x) of a finite permutation group G, which is the polynomial ∑ g∈G xn−π(g), where n is the degree of G and π(g) the number of fixed points of g ∈ G. In particular, we introduce a bivariate polynomial which is a special case of the cycle index of G, and from which ∆G(x) can be obtained, and then use this new polynomial to prove some identities relating the d...

Until 1980, there was no such subgroup as ‘infinite permutation groups’, according to the Mathematics Subject Classification: permutation groups were assumed to be finite. There were a few papers, for example [10, 62], and a set of lecture notes by Wielandt [72], from the 1950s. Now, however, there are far more papers on the topic than can possibly be summarised in an article like this one. I s...

We discuss some new properties of the natural Galois connection among set relation algebras, permutation groups, and first order logic. In particular, we exhibit infinitely many permutational relation algebras without a Galois closed representation, and we also show that every relation algebra on a set with at most six elements is Galois closed and essentially unique. Thus, we obtain the surpri...

A set of zero-one matrices satisfying (CC1)–(CC4) is called a coherent configuration. It is really a combinatorial object, since the conditions on the matrices can be translated into combinatorial conditions on the binary relations Oi. The coherent configuration formed by the orbital matrices of a permutation group G is the orbital configuration of G. Indeed, a coherent configuration is a parti...

For several families F of finite transitive permutation groups it is shown that each finite group is isomorphic to a 2-point stabilizer of infinitely many members of F .

The Hamming distance between two permutations of a finite set X is the number of elements of X on which they differ. In the first part of this paper, we consider bounds for the cardinality of a subset (or subgroup) of a permutation group P on X with prescribed distances between its elements. In the second part. We consider similar results for sets of s-tuples of permutations; the role of Hammin...