IBIS soluble linear groups

Let $G$ be a finite permutation group on $\Omega.$ An ordered sequence $(\omega_1,\dots, \omega_t)$ of elements of $\Omega$ is an irredundant base for $G$ if the pointwise stabilizer is trivial and no point is fixed by the stabilizer of its predecessors. If all irredundant bases of $G$ have the same cardinality, $G$ is said to be an IBIS group. In this paper we give a classification of quasi-primitive soluble irreducible IBIS linear groups, and we also describe nilpotent and metacyclic IBIS linear groups and IBIS linear groups of odd order.Comment: arXiv admin note: text overlap with arXiv:2206.01456 by other author

Permutation codes

AbstractThere are many analogies between subsets and permutations of a set, and in particular between sets of subsets and sets of permutations. The theories share many features, but there are also big differences. This paper is a survey of old and new results about sets (and groups) of permutations, concentrating on the analogies and on the relations to coding theory. Several open problems are described

Bases for permutation groups and matroids

AbstractIn this paper, we give two equivalent conditions for the irredundant bases of a permutation group to be the bases of a matroid. (These are deduced from a more general result for families of sets.) If they hold, then the group acts geometrically on the matroid, in the sense that the fixed points of any element form a flat. Some partial results towards a classification of such permutation groups are given. Further, if G acts geometrically on a perfect matroid design, there is a formula for the number of G-orbits on bases in terms of the cardinalities of flats and the numbers of G-orbits on tuples. This reduces, in a particular case, to the inversion formula for Stirling numbers