A transversal property for permutation groups motivated by partial transformations

João Araújo; João Pedro Araújo; Wolfram Bentz; Peter J. Cameron; Pablo Spiga

doi:10.1016/j.jalgebra.2020.12.024

A transversal property for permutation groups motivated by partial transformations

João Araújo^*, João Pedro Araújo, Wolfram Bentz, Peter J. Cameron, Pablo Spiga

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper we introduce the definition of the (k,l)-universal transversal property for permutation groups, which is a refinement of the definition of k-universal transversal property, which in turn is a refinement of the classical definition of k-homogeneity for permutation groups. In particular, a group possesses the (2,n)-universal transversal property if and only if it is primitive; it possesses the (2,2)-universal transversal property if and only if it is 2-homogeneous. Up to a few undecided cases, we give a classification of groups satisfying the (k,l)-universal transversal property, for k ≥ 3. Then we apply this result for studying regular semigroups of partial transformations.

Original language	English
Pages (from-to)	741-759
Number of pages	19
Journal	Journal of Algebra
Volume	573
Early online date	5 Jan 2021
DOIs	https://doi.org/10.1016/j.jalgebra.2020.12.024
Publication status	Published - 1 May 2021

Keywords

Primitive permutation groups

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10.1016/j.jalgebra.2020.12.024

Araújo-2021-A-transversal-property-for-JoA-AAM
Copyright © 2021 Elsevier Inc. All rights reserved. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1016/j.jalgebra.2020.12.024.
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title = "A transversal property for permutation groups motivated by partial transformations",

abstract = "In this paper we introduce the definition of the (k,l)-universal transversal property for permutation groups, which is a refinement of the definition of k-universal transversal property, which in turn is a refinement of the classical definition of k-homogeneity for permutation groups. In particular, a group possesses the (2,n)-universal transversal property if and only if it is primitive; it possesses the (2,2)-universal transversal property if and only if it is 2-homogeneous. Up to a few undecided cases, we give a classification of groups satisfying the (k,l)-universal transversal property, for k ≥ 3. Then we apply this result for studying regular semigroups of partial transformations.",

keywords = "Primitive permutation groups",

author = "Jo{\~a}o Ara{\'u}jo and Ara{\'u}jo, {Jo{\~a}o Pedro} and Wolfram Bentz and Cameron, {Peter J.} and Pablo Spiga",

note = "Funding: The first and third authors were partially supported by the Funda{\c c}{\~a}o para a Ci{\^e}ncia e a Tecnologia (Portuguese Foundation for Science and Technology) through projects UIDB/00297/2020 Center for Mathematics and Applications) and CEMAT-CI{\^E}NCIAS UID/Multi/04621/2013, and also by the Funda{\c c}{\~a}o para a Ci{\^e}ncia e a Tecnologia through project PTDC/MAT-PUR/31174/2017. The second author was partially supported by Funda{\c c}{\~a}o Calouste Gulbenkian, Programa Talentos Intelig{\^e}ncia Artificial. The fourth author was partially supported by Funda{\c c}{\~a}o para a Ci{\^e}ncia e a Tecnologia (Portuguese Foundation for Science and Technology) through project CEMAT-CI{\^E}NCIAS UID/Multi/04621/2013.",

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doi = "10.1016/j.jalgebra.2020.12.024",

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AU - Araújo, João

AU - Araújo, João Pedro

AU - Bentz, Wolfram

AU - Cameron, Peter J.

AU - Spiga, Pablo

N1 - Funding: The first and third authors were partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through projects UIDB/00297/2020 Center for Mathematics and Applications) and CEMAT-CIÊNCIAS UID/Multi/04621/2013, and also by the Fundação para a Ciência e a Tecnologia through project PTDC/MAT-PUR/31174/2017. The second author was partially supported by Fundação Calouste Gulbenkian, Programa Talentos Inteligência Artificial. The fourth author was partially supported by Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through project CEMAT-CIÊNCIAS UID/Multi/04621/2013.

PY - 2021/5/1

Y1 - 2021/5/1

N2 - In this paper we introduce the definition of the (k,l)-universal transversal property for permutation groups, which is a refinement of the definition of k-universal transversal property, which in turn is a refinement of the classical definition of k-homogeneity for permutation groups. In particular, a group possesses the (2,n)-universal transversal property if and only if it is primitive; it possesses the (2,2)-universal transversal property if and only if it is 2-homogeneous. Up to a few undecided cases, we give a classification of groups satisfying the (k,l)-universal transversal property, for k ≥ 3. Then we apply this result for studying regular semigroups of partial transformations.

AB - In this paper we introduce the definition of the (k,l)-universal transversal property for permutation groups, which is a refinement of the definition of k-universal transversal property, which in turn is a refinement of the classical definition of k-homogeneity for permutation groups. In particular, a group possesses the (2,n)-universal transversal property if and only if it is primitive; it possesses the (2,2)-universal transversal property if and only if it is 2-homogeneous. Up to a few undecided cases, we give a classification of groups satisfying the (k,l)-universal transversal property, for k ≥ 3. Then we apply this result for studying regular semigroups of partial transformations.

KW - Primitive permutation groups

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SP - 741

EP - 759

JO - Journal of Algebra

JF - Journal of Algebra

ER -

A transversal property for permutation groups motivated by partial transformations

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