An algorithm is given tjat recognises (in O(lN2 log N) time, where N is the size of the input and l the depth of a precalculated Schreier tree) when a transitive group, (G, Ω) is the action on one orbit of the action of G on the set Γ(2) of ordered pairs of distinct elements of some G-set Γ (that us, Ωis isomorphic to an orbital of (G,Γ)). This may be adapted to list all essentially different such actions in O(lN4log N)time, where N is the sum of sizes of the input and output. This will be a useful tool for reducing the degree of a permutation group as an aid to further study of the group.

This algorithm is then extended to provide an algorithm that will (in O(lN3 log N) time) recognise when a transiteve group is the action on one orbit of the action of G on the set Γ{2} ofunorderd pairs of distinct elements of some G-set Γ. An algorithm for finding all essentially different such actions is also provided, running in O(lN4logN) time. (again, N is the sum of the input and output sizes.) It is also indicated how these results may be applied to the more general problem of recognising when an intransitive group (G,Ω) is isomorphic to (G, Γ{2}) for some G-set Γ.

All the algorithms are practical; most have been implementd in GAP, and the code is made available with this paper. In some cases the algorithms are considerably more practical than their asymptotic analyses would suggest.

The paper gives formulae for a module presentation of the module of identities among relations for a presentation of a group, in terms of information on 0- and 1-combings of the Cayley graph. These formulae are seen as a special case of formulae for extending a partial free crossed resolution of a group, given a partial contracting homotopy of the universal cover of the partial resolution.

A reduction theory is developed for binary forms (homogeneous polynomials) of degrees three and four with integer coefficients. The resulting coefficient bounds simplify and improve on those in the literature, particularly in the case of negative discriminant. Applications include systematic enumeration of cubic number fields, and 2-descent on elliptic curves defined over the set of rational numbers. Remarks are given concerning the extension of these results to forms defined over number fields.

For each group G of order up to 30 we compute a small 3-dimensional CW-space X with π1X≌ G and π2X = 0, and we quantify the ‘efficiency’ of X. Furthermore, we give a theoretical result for treating the case when G is a semi-direct product of two groups for which 3-presentations are known. We also describe the ZG-module structure on the second homotopy group π2X2 of the 2-skeleton of X. This module structure can in principle be used to determine the co-homology groups H2(G, A) and H3(G, A) with coefficients in a ZG-module A. Our computations, which involve the Todd–Coxeter procedure for coset enumeration and the LLL algorithm for finding bases of integer lattices, are rather naive in that the LLL algorithm is applied to matrices of dimension a multiple of |G|. Thus, in their present form, our techniques can be used only on small groups (say of order up to several hundred). They can in principle be used to construct (crossed) ZG-resolutions of Z, but again, only for small G. The paper is accompanied by two attachment files. The first of these is a summary of our computations in HTML format. The second contains various GAP programs used in the computations.

Let E1 and E2 be ordinary elliptic curves over a finite field Fp such that #E1(Fp) = #E2(Fp). Tate's isogeny theorem states that there is an isogeny from E1 to E2 which is defined over Fp. The goal of this paper is to describe a probabilistic algorithm for constructing such an isogeny.

The algorithm proposed in this paper has exponential complexity in the worst case. Nevertheless, it is efficient in certain situations (that is, when the class number of the endomorphism ring is small). The significance of these results to elliptic curve cryptography is discussed.

The paper describes an approach to the computation of the zero energy thresholds for the appearance of negative energy eigenvalues of Schrödinger operators.