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The existential transversal property: a generalization of homogeneity and its impact on semigroups. (English) Zbl 07291895
Summary: Let $$G$$ be a permutation group of degree $$n$$, and $$k$$ a positive integer with $$k\le n$$. We say that $$G$$ has the $$k$$-existential transversal property, or $$k$$-et, if there exists a $$k$$-subset $$A$$ (of the domain $$\Omega )$$ whose orbit under $$G$$ contains transversals for all $$k$$-partitions $$\mathcal{P}$$ of $$\Omega$$. 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 $$\Omega$$.
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 $$k\ge 6$$ the only ones are alternating or symmetric groups; here we show that in the $$k$$-et context the threshold is 8, that is, for $$8\le k\le n/2$$, the only transitive groups with $$k$$-et are the symmetric and alternating groups; this is best possible since the Mathieu group $$M_{24}$$ (degree 24) has 7-et. We determine all groups with $$k$$-et for $$4\le k\le n/2$$, up to some unresolved cases for $$k=4,5$$, and describe the property for $$k=2,3$$ in permutation group language. These considerations essentially answer Problem 5 proposed in the paper on $$k$$-ut referred to above; we also slightly improve the classification of groups possessing the $$k$$-ut property.
In that earlier paper, the results were applied to semigroups, in particular, to the question of when the semigroup $$\langle G,t\rangle$$ is regular, where $$t$$ is a map of rank $$k$$ (with $$k<n/2)$$; this turned out to be equivalent to the $$k$$-ut property. The question investigated here is when there is a $$k$$-subset $$A$$ of the domain such that $$\langle G, t\rangle$$ is regular for all maps $$t$$ with image $$A$$. This turns out to be much more delicate; the $$k$$-et property (with $$A$$ as witnessing set) is a necessary condition, and the combination of $$k$$-et and $$(k-1)$$-ut is sufficient, but the truth lies somewhere between.
Given the knowledge that a group under consideration has the necessary condition of $$k$$-et, the regularity question for $$k\le n/2$$ is solved except for one sporadic group.
The paper ends with a number of problems on combinatorics, permutation groups and transformation semigroups, and their linear analogues.
##### MSC:
 20B30 Symmetric groups 20B35 Subgroups of symmetric groups 20B15 Primitive groups 20M20 Semigroups of transformations, relations, partitions, etc. 20M17 Regular semigroups
GAP
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