The classification of 3/2-transitive permutation groups and 1/2-transitive linear groups

A linear group G ≤ GL(V ), where V is a finite vector space, is called 12 -transitive if all the G-orbits on the set of nonzero vectors have the same size. We complete the classification of all the 12-transitive linear groups. As a consequence we complete the determination of the finite 32-transitive permutation groups – the transitive groups for which a point-stabilizer has all its nontrivial orbits of the same size. We also determine the (k +1 2)-transitive groups for integers k ≥ 2

Simple groups, product actions, and generalized quadrangles

The classification of flag-transitive generalized quadrangles is a long-standing open problem at the interface of finite geometry and permutation group theory. Given that all known flag-transitive generalized quadrangles are also point-primitive (up to point–line duality), it is likewise natural to seek a classification of the point-primitive examples. Working toward this aim, we are led to investigate generalized quadrangles that admit a collineation group$G$preserving a Cartesian product decomposition of the set of points. It is shown that, under a generic assumption on$G$, the number of factors of such a Cartesian product can be at most four. This result is then used to treat various types of primitive and quasiprimitive point actions. In particular, it is shown that$G$cannot haveholomorph compoundO’Nan–Scott type. Our arguments also pose purely group-theoretic questions about conjugacy classes in nonabelian finite simple groups and fixities of primitive permutation groups.</jats:p

A path-deformation framework for determining weighted genome rearrangement distance

Measuring the distance between two bacterial genomes under the inversion process is usually done by assuming all inversions to occur with equal probability. Recently, an approach to calculating inversion distance using group theory was introduced, and is effective for the model in which only very short inversions occur. In this paper, we show how to use the group-theoretic framework to establish minimal distance for any weighting on the set of inversions, generalizing previous approaches. To do this we use the theory of rewriting systems for groups, and exploit the Knuth--Bendix algorithm, the first time this theory has been introduced into genome rearrangement problems. The central idea of the approach is to use existing group theoretic methods to find an initial path between two genomes in genome space (for instance using only short inversions), and then to deform this path to optimality using a confluent system of rewriting rules generated by the Knuth--Bendix algorithm

Derangements and eigenvalue-free elements in finite classical groups

Permutations that have no fixed points have been known for a very long time as ‘derangements ’. Under that heading Rouse-Ball [10, p. 46] puts the matter in the following charming way: ‘Suppose you have written a letter to each of n different friends, and addressed the n corresponding envelopes. In how many ways can yo