This paper investigates the hardware implementation of arithmetical operations (multiplication and inversion) in symmetric and alternating groups, as well as in binary permutation groups (permutation groups of order 2). Various fast and space-efficient hardware architectures will be presented. High speed is achieved by employing switching networks, which effect multiplication in one clock cycle...

This article is about the mathematics of ringing the changes. We describe the mathematics which arises from a real-world activity, that of ringing the changes on bells. We present Rankin’s solution of one of the famous old problems in the subject.

We present new algorithms for permutation group manipulation. Our methods result in an improvement of nearly an order of magnitude in the worst-case analysis for the fundamental problems of finding strong generating sets and testing membership. The normal structure of the group is brought into play even for such elementary issues. An essential element is the recognition of large alternating com...

IN this note, we consider the following problem. Let G be a finite permutation group of degree d, and let N b e a normal subgroup of G. Under what circumstances does G/N have a faithful permutation representation of degree at most di Positive answers to this question are likely to have applications to computational group theory, since there are currently no really satisfactory practical methods...

Various descending chains of subgroups of a finite permutation group can be used to define a sequence of 'basic' permutation groups that are analogues of composition factors for abstract finite groups. Primitive groups have been the traditional choice for this purpose, but some combinatorial applications require different kinds of basic groups, such as quasiprimitive groups, that are defined by...

LetM(n, d) be the maximum size of a permutation array on n symbols with pairwise Hamming distance at least d. We use various combinatorial, algebraic, and computational methods to improve lower bounds forM(n, d). We compute the Hamming distances of affine semilinear groups and projective semilinear groups, and unions of cosets of AGL(1, q) and PGL(2, q) with Frobenius maps to obtain new, improv...

We investigate the combinatorics and geometry of permutation polytopes associated to cyclic permutation groups, i.e., the convex hulls of cyclic groups of permutation matrices. In the situation that the generator of the group consists of at most two orbits, we can give a complete combinatorial description of the associated permutation polytope. In the case of three orbits the facet structure is...

In collaboration with João Araújo and others, I have been thinking about relations between permutation groups and transformation semigroups. In particular, what properties of a permutation group G on a set Ω guarantee that, if f is any non-permutation on Ω (or perhaps any non-permutation of rank k), then the transformation semigroup 〈G, f〉 has specified properties. There is far too much to summ...

A remarkable fact, discovered by Wang in [14], is that the set Xn = {1, . . . , n} has a quantum permutation group. For n = 1, 2, 3 this is the usual symmetric group Sn. However, starting from n = 4 the situation is different: for instance the dual of Z2 ∗ Z2 acts on X4. In other words, “quantum permutations” do exist. They form a compact quantum group Qn, satisfying the axioms of Woronowicz in...

For G ≤ Sym(Ω), let π(G) be the set of partitions of Ω given by the cycles of elements of G. Under the refinement order, π(G) admits join and meet operations. We say thatG is joinor meet-coherent if π(G) is joinor meet-closed, respectively. The centralizer in Sym(Ω) of any permutation g is shown to be meetcoherent, and join-coherent subject to a finiteness condition. Hence if G is a centralizer...