A new notion of transitivity for groups and sets of permutations
| dc.creator | Martin, William J. | |
| dc.creator | Sagan, Bruce E. | |
| dc.date | 2002-06-17 | |
| dc.date.accessioned | 2026-07-07T04:49:09Z | |
| dc.date.available | 2026-07-07T04:49:09Z | |
| dc.description | Let $Ω=\{1,2,...,n\}$ where $n \ge 2$. The {\em shape} of an ordered set partition $P=(P_1,..., P_k)$ of $Ω$ is the integer partition $λ=(λ_1,...,λ_k)$ defined by $λ_i = |P_i|$. Let G be a group of permutations acting on $Ω$. For a fixed partition $λ$ of n, we say that G is {\em $λ$-transitive} if G has only one orbit when acting on partitions P of shape $\la$. A corresponding definition can also be given when G is just a set. For example, if $λ=(n-t,1,...,1)$, then a $λ$-transitive group is the same as a t-transitive permutation group and if $λ=(n-t,t)$, then we recover the t-homogeneous permutation groups. In this paper, we use the character theory of the symmetric group $S_n$ to establish some structural results regarding $λ$-transitive groups and sets. In particular, we are able to generalize a theorem of Livingstone and Wagner 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 non-trivial t-transitive set of permutations for each positive integer t. We also show how these ideas lead to a split basis for the association scheme of the symmetric group. | |
| dc.description | 13 pages | |
| dc.identifier | https://arxiv.org/abs/math/0206163 | |
| dc.identifier | http://arxiv.org/abs/math/0206163 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/64317 | |
| dc.subject | Group Theory | |
| dc.subject | Combinatorics | |
| dc.subject | 20B20 (Primary) 20C30, 05E10, 05E30 (Secondary) | |
| dc.title | A new notion of transitivity for groups and sets of permutations | |
| dc.type | text |