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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
Software:
GAP
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References:
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