The existential transversal property: A generalization of homogeneity and its impact on semigroups

Abstract

Let G be a permutation group of degree n, and k a positive integer with k ≤ n. We say that G has the k-existential transversal property, or k-et, if there exists a k-subset A (of the domain Ω) whose orbit under G contains transversals for all k-partitions P of Ω. This property is a substantial weakening of the k-universal transversal property, or k-ut, investigated by the first and third author, which required this condition to hold for all k-subsets A of the domain Ω. Our first task in this paper is to investigate the k-et property and to decide which groups satisfy it. For example, it is known that for k < 6 there are several families of k-transitive groups, but for k ≥ 6 the only ones are alternating or symmetric groups; here we show that in the k-et context the threshold is 8, that is, for 8 ≤ k ≤ n/2, the only transitive groups with k-et are the symmetric and alternating

A transversal property for permutation groups motivated by partial transformations

“…On the other hand, the results of [3,8] give a nearly complete classification of groups with the k-ut property for 3 ≤ k ≤ n − 2, and so it is entirely reasonable to ask for a similar classification of groups satisfying (k, l)-ut for any such k and all l with k ≤ l ≤ n; and this we do (up to a few unresolved cases). The results are summarised below.…”

“…Let A ⊆ T (Ω), the full transformation monoid on Ω; classify the permutation groups G ≤ S n such that G, a is regular for all a ∈ A. For many different sets A, this problem has been considered in [2,3,4,8,10,19,20,21,25], among others. The goal of this paper is to consider the similar problem when A is a set of partial transformations with prescribed domain and image sizes (Theorem 1.6).…”

“…The groups satisfying 3-ut were partially classified in [8]; in [3], these results were extended and some minor corrections added to obtain the following classification. Proposition 6.1.…”

“…Proof of Theorem 1.4. From [3], we see that either G is 4-homogeneous (that is, t(G, 4) = 4), or (n, G) is one of the four cases listed in the statement of Theorem 1.4. For the first three cases, the value of t(G, 4) follows with a computation.…”

“…A degree n permutation group on Ω is said to have the k-et property if there exists some k-set K ⊆ Ω such that the orbit of K under G contains transversals for every k-partition of Ω [3]. The k-et property generalizes the k-ut property.…”