Classification of designs with nontrivial automorphism groups

Abstract

In this article, we introduce a new orderly backtrack algorithm with efficient isomorph rejection for classification of t-designs. As an application, we classify all simple 2-ð13; 3; 2Þ designs with nontrivial automorphism groups. The total number of such designs amounts to 1; 897; 386. The decomposability of the designs is also considered.

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

“…When classifying H-invariant objects up to G-equivalence (that is, up to being in the same G-orbit), for a given H ≤ G, one often first classifies the objects up to N G (H)-equivalence (see [1], [15]). Proposition 2.1 allows us to avoid many tests to determine G-equivalence when N G (H)-orbit representatives of the H-invariant objects have already been determined.…”

“…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}).…”

The Mendelsohn Triple Systems of Order 13

“…In both of these cases, one may utilize Theorem and restrict to objects with certain symmetries of order 2. A partial classification of TTS(13) with nontrivial automorphisms in shows that there are more than a million such objects with an automorphism of order 2. Theorem The TTS underlying a self‐converse MTS either consists of two copies of a Steiner triple system or has an automorphism of order 2. Proof If an MTS on the point set V is self‐converse, then there is a bijection from V to V —which we view as a permutation α—that maps the converse of onto .…”