Orbits of primitive $k$-homogenous groups on $(n-k)$-partitions with applications to semigroups

Abstract

Let X be a finite set such that |X| = n, and let k < n/2. A group is k-homogeneous if it has only one orbit on the sets of size k. The aim of this paper is to prove some general results on permutation groups and then apply them to transformation semigroups. On groups we find the minimum number of permutations needed to generate k-homogeneous groups (for k ≥ 1); in particular we show that 2-homogeneous groups are 2-generated. We also describe the orbits of k-homogenous groups on partitions with n − k parts, classify the 3-homogeneous groups G whose orbits on (n − 3)-partitions are invariant under the normalizer of G in S n , and describe the normalizers of 2-homogeneous groups in the symmetric group. Then these results are applied to extract information about transformation semigroups with given group of units, namely to prove results on their automorphisms and on the minimum number of generators. The paper finishes with some problems on permutation groups, transformation semigroups and computational algebra.

A transversal property for permutation groups motivated by partial transformations

“…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).…”

“…Proof of Theorem 1.3. From [8, Theorem 1.5], we see that either G is 5-homogeneous (that is, t(G, 5) = 5), or n = 33 and G = PΓL (2,32). In the second case, the determination of t(G, 5) follows with a computation.…”

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

“…For G with socle PSU(3, q), 2 B 2 (q), or 2 G 2 (q), n ≥ 33, the order bound fails except for PΓU (3,4), k = 5. This case can be excluded by having too many orbits on 4-sets.…”

“…If any other groups G are 4-et, then they satisfy one of the following: In the last case, note that there are 3 non-isomorphic groups of the form PSU (3,8).3. Only one of those has less than 5 orbits on 3-sets and could be 4-et.…”

Imprimitive permutations in primitive groups

“…By Corollary 2.4 of [4], every 2-homogeneous group G has d(G) = 2 and hence to compute m(n) we only have to consider the other primitive groups. We observe that the bound cn √ log n is the best possible apart from constants; examples are in [24].…”