Pub Date : 2021-03-03DOI: 10.26493/1855-3974.1840.6E0
G. Jones
It is shown that in various categories, including many consisting of maps or hypermaps, oriented or unoriented, of a given hyperbolic type, or of coverings of a suitable topological space, every countable group A is isomorphic to the automorphism group of uncountably many non-isomorphic objects, infinitely many of them finite if A is finite. In particular, the latter applies to dessins d’enfants, regarded as finite oriented hypermaps.
{"title":"Realisation of groups as automorphism groups in permutational categories","authors":"G. Jones","doi":"10.26493/1855-3974.1840.6E0","DOIUrl":"https://doi.org/10.26493/1855-3974.1840.6E0","url":null,"abstract":"It is shown that in various categories, including many consisting of maps or hypermaps, oriented or unoriented, of a given hyperbolic type, or of coverings of a suitable topological space, every countable group A is isomorphic to the automorphism group of uncountably many non-isomorphic objects, infinitely many of them finite if A is finite. In particular, the latter applies to dessins d’enfants, regarded as finite oriented hypermaps.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90414156","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-22DOI: 10.26493/1855-3974.2568.55c
Aleksander Kelenc, Aoden Teo Masa Toshi, R. Škrekovski, I. Yero
The metric (resp. edge metric or mixed metric) dimension of a graph $G$, is the cardinality of the smallest ordered set of vertices that uniquely recognizes all the pairs of distinct vertices (resp. edges, or vertices and edges) of $G$ by using a vector of distances to this set. In this note we show two unexpected results on hypercube graphs. First, we show that the metric and edge metric dimension of $Q_d$ differ by only one for every integer $d$. In particular, if $d$ is odd, then the metric and edge metric dimensions of $Q_d$ are equal. Second, we prove that the metric and mixed metric dimensions of the hypercube $Q_d$ are equal for every $d ge 3$. We conclude the paper by conjecturing that all these three types of metric dimensions of $Q_d$ are equal when $d$ is large enough.
{"title":"On metric dimensions of hypercubes","authors":"Aleksander Kelenc, Aoden Teo Masa Toshi, R. Škrekovski, I. Yero","doi":"10.26493/1855-3974.2568.55c","DOIUrl":"https://doi.org/10.26493/1855-3974.2568.55c","url":null,"abstract":"The metric (resp. edge metric or mixed metric) dimension of a graph $G$, is the cardinality of the smallest ordered set of vertices that uniquely recognizes all the pairs of distinct vertices (resp. edges, or vertices and edges) of $G$ by using a vector of distances to this set. In this note we show two unexpected results on hypercube graphs. First, we show that the metric and edge metric dimension of $Q_d$ differ by only one for every integer $d$. In particular, if $d$ is odd, then the metric and edge metric dimensions of $Q_d$ are equal. Second, we prove that the metric and mixed metric dimensions of the hypercube $Q_d$ are equal for every $d ge 3$. We conclude the paper by conjecturing that all these three types of metric dimensions of $Q_d$ are equal when $d$ is large enough.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85122371","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-22DOI: 10.26493/1855-3974.2593.1B7
Junzhi Huang, Yan-Quan Feng, Jin-Xin Zhou
In this paper, we construct an infinite family of normal Cayley graphs, which are 2 -distance-transitive but neither distance-transitive nor 2 -arc-transitive. This answers a question proposed by Chen, Jin and Li in 2019.
{"title":"Two-distance transitive normal Cayley graphs","authors":"Junzhi Huang, Yan-Quan Feng, Jin-Xin Zhou","doi":"10.26493/1855-3974.2593.1B7","DOIUrl":"https://doi.org/10.26493/1855-3974.2593.1B7","url":null,"abstract":"In this paper, we construct an infinite family of normal Cayley graphs, which are 2 -distance-transitive but neither distance-transitive nor 2 -arc-transitive. This answers a question proposed by Chen, Jin and Li in 2019.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"28 1","pages":"2"},"PeriodicalIF":0.0,"publicationDate":"2021-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73466677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-15DOI: 10.26493/1855-3974.2558.173
Gabriel Verret, Binzhou Xia
In this paper, we show that every finite simple group of order at least 5 admits an oriented regular representation of out-valency 2.
在本文中，我们证明了每一个至少5阶的有限单群都有一个外价2的有向正则表示。
{"title":"Oriented regular representations of out-valency two for finite simple groups","authors":"Gabriel Verret, Binzhou Xia","doi":"10.26493/1855-3974.2558.173","DOIUrl":"https://doi.org/10.26493/1855-3974.2558.173","url":null,"abstract":"In this paper, we show that every finite simple group of order at least 5 admits an oriented regular representation of out-valency 2.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"10 1","pages":"1"},"PeriodicalIF":0.0,"publicationDate":"2021-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74913214","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-10DOI: 10.26493/1855-3974.2554.856
A. S. Razafimahatratra
Given a finite transitive permutation group $Gleq operatorname{Sym}(Omega)$, with $|Omega|geq 2$, the derangement graph $Gamma_G$ of $G$ is the Cayley graph $operatorname{Cay}(G,operatorname{Der}(G))$, where $operatorname{Der}(G)$ is the set of all derangements of $G$. Meagher et al. [On triangles in derangement graphs, J. Combin. Theory, Ser. A, 180:105390, 2021] recently proved that $operatorname{Sym}(2)$ acting on ${1,2}$ is the only transitive group whose derangement graph is bipartite and any transitive group of degree at least three has a triangle in its derangement graph. They also showed that there exist transitive groups whose derangement graphs are complete multipartite. This paper gives two new families of transitive groups with complete multipartite derangement graphs. In addition, we construct an infinite family of transitive groups whose derangement graphs are multipartite but not complete.
{"title":"On complete multipartite derangement graphs","authors":"A. S. Razafimahatratra","doi":"10.26493/1855-3974.2554.856","DOIUrl":"https://doi.org/10.26493/1855-3974.2554.856","url":null,"abstract":"Given a finite transitive permutation group $Gleq operatorname{Sym}(Omega)$, with $|Omega|geq 2$, the derangement graph $Gamma_G$ of $G$ is the Cayley graph $operatorname{Cay}(G,operatorname{Der}(G))$, where $operatorname{Der}(G)$ is the set of all derangements of $G$. Meagher et al. [On triangles in derangement graphs, J. Combin. Theory, Ser. A, 180:105390, 2021] recently proved that $operatorname{Sym}(2)$ acting on ${1,2}$ is the only transitive group whose derangement graph is bipartite and any transitive group of degree at least three has a triangle in its derangement graph. They also showed that there exist transitive groups whose derangement graphs are complete multipartite. \u0000This paper gives two new families of transitive groups with complete multipartite derangement graphs. In addition, we construct an infinite family of transitive groups whose derangement graphs are multipartite but not complete.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91382743","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-06DOI: 10.26493/1855-3974.2550.d96
M'at'e Kadlicsk'o, Z. L'angi
The concept of a Minkowski arrangement was introduced by Fejes T'oth in 1965 as a family of centrally symmetric convex bodies with the property that no member of the family contains the center of any other member in its interior. This notion was generalized by Fejes T'oth in 1967, who called a family of centrally symmetric convex bodies a generalized Minkowski arrangement of order $mu$ for some $0
闵可夫斯基排列的概念是由Fejes T oth在1965年引入的，它是一组中心对称的凸体，其性质是该族中的任何成员都不包含其内部任何其他成员的中心。这个概念在1967年由Fejes T 'oth推广，他将中心对称凸体族称为阶$mu$的广义闵可夫斯基排列，对于某些$0
{"title":"On generalized Minkowski arrangements","authors":"M'at'e Kadlicsk'o, Z. L'angi","doi":"10.26493/1855-3974.2550.d96","DOIUrl":"https://doi.org/10.26493/1855-3974.2550.d96","url":null,"abstract":"The concept of a Minkowski arrangement was introduced by Fejes T'oth in 1965 as a family of centrally symmetric convex bodies with the property that no member of the family contains the center of any other member in its interior. This notion was generalized by Fejes T'oth in 1967, who called a family of centrally symmetric convex bodies a generalized Minkowski arrangement of order $mu$ for some $0<mu<1$ if no member $K$ of the family overlaps the homothetic copy of any other member $K'$ with ratio $mu$ and with the same center as $K'$. In this note we prove a sharp upper bound on the total area of the elements of a generalized Minkowski arrangement of order $mu$ of finitely many circular disks in the Euclidean plane. This result is a common generalization of a similar result of Fejes T'oth for Minkowski arrangements of circular disks, and a result of B\"or\"oczky and Szab'o about the maximum density of a generalized Minkowski arrangement of circular disks in the plane. In addition, we give a sharp upper bound on the density of a generalized Minkowski arrangement of homothetic copies of a centrally symmetric convex body.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87761883","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-05DOI: 10.26493/1855-3974.2621.26f
S. Mirafzal
Let $Gamma=(V,E)$ be a graph. The square graph $Gamma^2$ of the graph $Gamma$ is the graph with the vertex set $V(Gamma^2)=V$ in which two vertices are adjacent if and only if their distance in $Gamma$ is at most two. The square graph of the hypercube $Q_n$ has some interesting properties. For instance, it is highly symmetric and panconnected. In this paper, we investigate some algebraic properties of the graph ${Q^2_n}$. In particular, we show that the graph ${Q^2_n}$ is distance-transitive. We show that the graph ${Q^2_n}$ is an imprimitive distance-transitive graph if and only if $n$ is an odd integer. Also, we determine the spectrum of the graph $Q_n^2$. Finally, we show that when $n>2$ is an even integer, then ${Q^2_n}$ is an automorphic graph, that is, $Q_n^2$ is a distance-transitive primitive graph which is not a complete or a line graph.
{"title":"Some remarks on the square graph of the hypercube","authors":"S. Mirafzal","doi":"10.26493/1855-3974.2621.26f","DOIUrl":"https://doi.org/10.26493/1855-3974.2621.26f","url":null,"abstract":"Let $Gamma=(V,E)$ be a graph. The square graph $Gamma^2$ of the graph $Gamma$ is the graph with the vertex set $V(Gamma^2)=V$ in which two vertices are adjacent if and only if their distance in $Gamma$ is at most two. The square graph of the hypercube $Q_n$ has some interesting properties. For instance, it is highly symmetric and panconnected. In this paper, we investigate some algebraic properties of the graph ${Q^2_n}$. In particular, we show that the graph ${Q^2_n}$ is distance-transitive. We show that the graph ${Q^2_n}$ is an imprimitive distance-transitive graph if and only if $n$ is an odd integer. Also, we determine the spectrum of the graph $Q_n^2$. Finally, we show that when $n>2$ is an even integer, then ${Q^2_n}$ is an automorphic graph, that is, $Q_n^2$ is a distance-transitive primitive graph which is not a complete or a line graph.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73761572","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-10DOI: 10.26493/1855-3974.2577.25d
L. K. Jørgensen, Guillermo Pineda-Villavicencio, J. Ugon
This paper is concerned with the linkedness of Cartesian products of complete graphs. A graph with at least $2k$ vertices is {it $k$-linked} if, for every set of $2k$ distinct vertices organised in arbitrary $k$ pairs of vertices, there are $k$ vertex-disjoint paths joining the vertices in the pairs. We show that the Cartesian product $K^{d_{1}+1}times K^{d_{2}+1}$ of complete graphs $K^{d_{1}+1}$ and $K^{d_{2}+1}$ is $floor{(d_{1}+d_{2})/2}$-linked for $d_{1},d_{2}ge 2$, and this is best possible. %A polytope is said to be {it $k$-linked} if its graph is $k$-linked. This result is connected to graphs of simple polytopes. The Cartesian product $K^{d_{1}+1}times K^{d_{2}+1}$ is the graph of the Cartesian product $T(d_{1})times T(d_{2})$ of a $d_{1}$-dimensional simplex $T(d_{1})$ and a $d_{2}$-dimensional simplex $T(d_{2})$. And the polytope $T(d_{1})times T(d_{2})$ is a {it simple polytope}, a $(d_{1}+d_{2})$-dimensional polytope in which every vertex is incident to exactly $d_{1}+d_{2}$ edges. While not every $d$-polytope is $floor{d/2}$-linked, it may be conjectured that every simple $d$-polytope is. Our result implies the veracity of the revised conjecture for Cartesian products of two simplices.
{"title":"Linkedness of Cartesian products of complete graphs","authors":"L. K. Jørgensen, Guillermo Pineda-Villavicencio, J. Ugon","doi":"10.26493/1855-3974.2577.25d","DOIUrl":"https://doi.org/10.26493/1855-3974.2577.25d","url":null,"abstract":"This paper is concerned with the linkedness of Cartesian products of complete graphs. A graph with at least $2k$ vertices is {it $k$-linked} if, for every set of $2k$ distinct vertices organised in arbitrary $k$ pairs of vertices, there are $k$ vertex-disjoint paths joining the vertices in the pairs. \u0000We show that the Cartesian product $K^{d_{1}+1}times K^{d_{2}+1}$ of complete graphs $K^{d_{1}+1}$ and $K^{d_{2}+1}$ is $floor{(d_{1}+d_{2})/2}$-linked for $d_{1},d_{2}ge 2$, and this is best possible. \u0000%A polytope is said to be {it $k$-linked} if its graph is $k$-linked. \u0000This result is connected to graphs of simple polytopes. The Cartesian product $K^{d_{1}+1}times K^{d_{2}+1}$ is the graph of the Cartesian product $T(d_{1})times T(d_{2})$ of a $d_{1}$-dimensional simplex $T(d_{1})$ and a $d_{2}$-dimensional simplex $T(d_{2})$. And the polytope $T(d_{1})times T(d_{2})$ is a {it simple polytope}, a $(d_{1}+d_{2})$-dimensional polytope in which every vertex is incident to exactly $d_{1}+d_{2}$ edges. \u0000While not every $d$-polytope is $floor{d/2}$-linked, it may be conjectured that every simple $d$-polytope is. Our result implies the veracity of the revised conjecture for Cartesian products of two simplices.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"61 1","pages":"2"},"PeriodicalIF":0.0,"publicationDate":"2020-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86056326","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-08DOI: 10.26493/978-961-293-027-1
Jonathan Leech
About the book: The extended study of non-commutative lattices was begun in 1949 by Ernst Pascual Jordan, a theoretical and mathematical physicist and co-worker of Max Born and Werner Karl Heisenberg. Jordan introduced noncommutative lattices as algebraic structures potentially suitable to encompass the logic of the quantum world. The modern theory of noncommutative lattices began forty years later with Jonathan Leech’s 1989 paper “Skew lattices in rings.” Recently, noncommutative generalizations of lattices and related structures have seen an upsurge in interest, with new ideas and applications emerging, from quasilattices to skew Heyting algebras. Much of this activity is derived in some way from the initiation of Jonathan Leech’s program of research in this area. The present book consists of seven chapters, mainly covering skew lattices, quasilattices and paralattices, skew lattices of idempotents in rings and skew Boolean algebras. As such, it is the first research monograph covering major results due to this renewed study of noncommutative lattices. It will serve as a valuable graduate textbook on the subject, as well as a handy reference to researchers of noncommutative algebras.
本书简介:非交换格的扩展研究始于1949年，由理论和数学物理学家恩斯特·帕斯夸尔·乔丹(Ernst Pascual Jordan)开始，他是马克斯·波恩(Max Born)和维尔纳·卡尔·海森堡(Werner Karl Heisenberg)的同事。Jordan引入了非交换格作为可能适合包含量子世界逻辑的代数结构。非交换格的现代理论始于四十年后的1989年Jonathan Leech的论文《环中的偏格》。近年来，格和相关结构的非交换推广引起了人们的极大兴趣，从拟格到斜Heyting代数，新的思想和应用不断涌现。这种活动在某种程度上源于乔纳森·里奇在这一领域的研究计划的启动。本书共分七章，主要涵盖了斜格、拟格和拟格、环中幂等的斜格和斜布尔代数。因此，它是第一个研究专著，涵盖了由于这一新研究的非交换格的主要结果。它将作为一个有价值的研究生教科书的主题，以及一个方便的参考研究非交换代数。
{"title":"Jonathan E. Leech: Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond","authors":"Jonathan Leech","doi":"10.26493/978-961-293-027-1","DOIUrl":"https://doi.org/10.26493/978-961-293-027-1","url":null,"abstract":"About the book: The extended study of non-commutative lattices was begun in 1949 by Ernst Pascual Jordan, a theoretical and mathematical physicist and co-worker of Max Born and Werner Karl Heisenberg. Jordan introduced noncommutative lattices as algebraic structures potentially suitable to encompass the logic of the quantum world. The modern theory of noncommutative lattices began forty years later with Jonathan Leech’s 1989 paper “Skew lattices in rings.” Recently, noncommutative generalizations of lattices and related structures have seen an upsurge in interest, with new ideas and applications emerging, from quasilattices to skew Heyting algebras. Much of this activity is derived in some way from the initiation of Jonathan Leech’s program of research in this area. The present book consists of seven chapters, mainly covering skew lattices, quasilattices and paralattices, skew lattices of idempotents in rings and skew Boolean algebras. As such, it is the first research monograph covering major results due to this renewed study of noncommutative lattices. It will serve as a valuable graduate textbook on the subject, as well as a handy reference to researchers of noncommutative algebras.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"109 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77056454","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-08DOI: 10.26493/1855-3974.2049.3DB
S. Glasby, Emilio Pierro, C. Praeger
We discuss recent progress on the problem of classifying point-primitive generalised polygons. In the case of generalised hexagons and generalised octagons, this has reduced the problem to primitive actions of almost simple groups of Lie type. To illustrate how the natural geometry of these groups may be used in this study, we show that if $mathcal{S}$ is a finite thick generalised hexagon or octagon with $G leqslant{rm Aut}(mathcal{S})$ acting point-primitively and the socle of $G$ isomorphic to ${rm PSL}_n(q)$ where $n geqslant 2$, then the stabiliser of a point acts irreducibly on the natural module. We describe a strategy to prove that such a generalised hexagon or octagon $mathcal{S}$ does not exist.
{"title":"Point-primitive generalised hexagons and octagons and projective linear groups","authors":"S. Glasby, Emilio Pierro, C. Praeger","doi":"10.26493/1855-3974.2049.3DB","DOIUrl":"https://doi.org/10.26493/1855-3974.2049.3DB","url":null,"abstract":"We discuss recent progress on the problem of classifying point-primitive generalised polygons. In the case of generalised hexagons and generalised octagons, this has reduced the problem to primitive actions of almost simple groups of Lie type. To illustrate how the natural geometry of these groups may be used in this study, we show that if $mathcal{S}$ is a finite thick generalised hexagon or octagon with $G leqslant{rm Aut}(mathcal{S})$ acting point-primitively and the socle of $G$ isomorphic to ${rm PSL}_n(q)$ where $n geqslant 2$, then the stabiliser of a point acts irreducibly on the natural module. We describe a strategy to prove that such a generalised hexagon or octagon $mathcal{S}$ does not exist.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79212770","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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