A NEW NOTION OF TRANSITIVITY FOR GROUPS AND SETS OF PERMUTATIONS

Abstract

Let $\Omega=\{1,2,\dotsc,n\}$ where $n \ge 2$. The shape of an ordered set partition $P=(P_1,\dotsc, P_k)$ of $\Omega$ is the integer partition $\lambda=(\lambda_1,\dotsc,\lambda_k)$ defined by $\lambda_i = |P_i|$. Let G be a group of permutations acting on $\Omega$. For a fixed partition $\lambda$ of n, we say that G is $\lambda$-transitive if G has only one orbit when acting on partitions P of shape $\lambda$. A corresponding definition can also be given when G is just a set. For example, if $\lambda=(n-t,1,\dotsc,1)$, then a $\lambda$-transitive group is the same as a t-transitive permutation group, and if $\lambda=(n-t,t)$, then we recover the t-homogeneous permutation groups.

We use the character theory of the symmetric group S_n to establish some structural results regarding $\lambda$-transitive groups and sets. In particular, we are able to generalize a celebrated result of Livingstone and Wagner [Math. Z. 90 (1965) 393–403] about t-homogeneous groups. We survey the relevant examples coming from groups. While it is known that a finite group of permutations can be at most 5-transitive unless it contains the alternating group, we show that it is possible to construct a nontrivial t-transitive set of permutations for each positive integer t. We also show how these ideas lead to a combinatorial basis for the Bose–Mesner algebra of the association scheme of the symmetric group and a design system attached to this association scheme.