# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## The existential transversal property: A generalization of homogeneity and its impact on semigroupsHTML articles powered by AMS MathViewer

by João Araújo, Wolfram Bentz and Peter J. Cameron
Trans. Amer. Math. Soc. 374 (2021), 1155-1195 Request permission

## Abstract:

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.

References
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• João Araújo
• Affiliation: Department of Mathematics and Centre for Mathematics and Applications (CMA), Faculdade de Ciências e Tecnologia (FCT) Universidade Nova de Lisboa (UNL), 2829-516 Caparica, Portugal; and CEMAT-Ciências, Faculdade de Ciências, Universidade de Lisboa 1749–016, Lisboa, Portugal
• MR Author ID: 664908
• Email: jj.araujo@fct.unl.pt, jjaraujo@fc.ul.pt
• Wolfram Bentz
• Affiliation: Centre for Mathematics and Applications (CMA), Faculdade de Ciêcias e Tecnologia (FCT), Universidade Nova de Lisboa (UNL), 2829-516, Caparica, Portugal; and Universidade Aberta, R. Escola Politécnica, 147, 1269-001 Lisboa, Portugal; and Department of Physics and Mathematics, University of Hull, Kingston upon Hull, HU6 7RX, United Kingdom
• MR Author ID: 641226
• Email: Wolfram.Bentz@uab.pt
• Peter J. Cameron
• Affiliation: School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, United Kingdom
• MR Author ID: 44560
• ORCID: 0000-0003-3130-9505
• Email: pjc20@st-andrews.ac.uk
• Received by editor(s): February 18, 2019
• Received by editor(s) in revised form: February 26, 2020, and May 26, 2020
• Published electronically: December 3, 2020
• Additional Notes: The corresponding author is W. Bentz
The first author was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the projects UIDB/00297/2020 (Centro de Matemáica e Aplicações), PTDC/MAT-PUR/31174/2017, UIDB/04621/2020 and UIDP/04621/2020.
The second author was supported by travel grants from the University of Hull’s Faculty of Science and Engineering and the Center for Computational and Stochastic Mathematics.