Minimal and canonical images

Abstract

We describe a family of new algorithms for finding the canonical image of a set of points under the action of a permutation group. This family of algorithms makes use of the orbit structure of the group, and a chain of subgroups of the group, to efficiently reduce the amount of search that must be performed to find a canonical image.We present a formal proof of correctness of our algorithms and describe experiments on different permutation groups that compare our algorithms with the previous state of the art.

Strong external difference families in abelian and non-abelian groups

“…Preprocessing of the group information was performed using GAP [3], in particular its SmallGroups library, while the search for SEDFs was implemented in Java, using a recursive depth-first search algorithm. The algorithm returns all (n, m, k, λ)-SEDFs in a group G; a final list of non-equivalent SEDFs is then produced using the Images package [7] in GAP. For abelian groups, results from the literature rule-out the following parameter sets: (9, 3, 2, 1), (10, 3, 3, 2), (13, 4, 2, 1), (17, 3, 4, 2), (17, 4, 4, 3), (17, 5, 2, 1), (19, 3, 3, 1), (19, 3, 6, 4), (19,5,3,2) [10]; (21, 6, 2, 1) (Proposition 2.6); (19, 2, 6, 2), (21, 2, 10, 5) [6].…”

On classifying objects with specified groups of automorphisms, friendly subgroups, and Sylow tower groups

“…For example, in GAP, when G is a permutation group, determining stabilisers of point-sets is faster in practice than Linton's important procedure [12] to find the lexicographically least set in a G-orbit of a given point-set. Although methods more efficient than Linton's are being developed [8], they still appear to require more work than determining a G-stabiliser of the set under consideration. On the other hand, there appears to be little time difference to computing the automorphism group of a graph and also canonically labelling that graph using nauty [13].…”

“…To illustrate the application of friendly subgroups, we now describe the classification of the maximal partial spreads invariant under a group of order 5 in both PG(3, 7) and PG (3,8). Some of the explanation below is taken from the author's webpage [17].…”

“…Since a Sylow 5-subgroup of G := PGL(4, 8) is also cyclic (in fact it is cyclic of order 5), we may classify the maximal partial spreads in PG (3,8) invariant under a group H of order 5 as we did in PG (3,7). We find that, up to N G (H)-equivalence, and hence up to G-equivalence, there are exactly (and no others).…”

“…Often a complete classification is too difficult, and we impose the additional condition that each object we seek is invariant under at least one of some specified (non-conjugate) subgroups of G. This is the situation we consider in this paper (see also [1], [4], [9], [10], [11], [14]). We are especially interested in the avoidance of unnecessary G-equivalence checks of H-invariant objects (which are often done via the determination of canonical G-orbit representatives of the given objects (see [8], [12], [13])), when often easier N G (H)-equivalence checks suffice (where N G (H) = {x ∈ G : x −1 Hx = H}).…”