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7-location, weak systolicity and isoperimetry 7-位置、收缩力弱和等压测量
arXiv - MATH - Group Theory Pub Date : 2024-09-01 DOI: arxiv-2409.00612
Nima Hoda, Ioana-Claudia Lazăr
{"title":"7-location, weak systolicity and isoperimetry","authors":"Nima Hoda, Ioana-Claudia Lazăr","doi":"arxiv-2409.00612","DOIUrl":"https://doi.org/arxiv-2409.00612","url":null,"abstract":"$m$-location is a local combinatorial condition for flag simplicial complexes\u0000introduced by Osajda. Osajda showed that simply connected 8-located locally\u00005-large complexes are hyperbolic. We treat the nonpositive curvature case of\u00007-located locally 5-large complexes. We show that any minimal area disc diagram in a 7-located locally 5-large\u0000complex is itself 7-located and locally 5-large. We define a natural CAT(0)\u0000metric for 7-located disc diagrams and use this to prove that simply connected\u00007-located locally 5-large complexes have quadratic isoperimetric function.\u0000Along the way, we prove that locally weakly systolic complexes are 7-located\u0000locally 5-large.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206478","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}
引用次数: 0
A classification of finite groups with small Davenport constant 具有小达文波特常数的有限群分类
arXiv - MATH - Group Theory Pub Date : 2024-08-31 DOI: arxiv-2409.00363
Jun Seok Oh
{"title":"A classification of finite groups with small Davenport constant","authors":"Jun Seok Oh","doi":"arxiv-2409.00363","DOIUrl":"https://doi.org/arxiv-2409.00363","url":null,"abstract":"Let $G$ be a finite group. By a sequence over $G$, we mean a finite unordered\u0000string of terms from $G$ with repetition allowed, and we say that it is a\u0000product-one sequence if its terms can be ordered so that their product is the\u0000identity element of $G$. Then, the Davenport constant $mathsf D (G)$ is the\u0000maximal length of a minimal product-one sequence, that is a product-one\u0000sequence which cannot be partitioned into two non-trivial product-one\u0000subsequences. The Davenport constant is a combinatorial group invariant that\u0000has been studied fruitfully over several decades in additive combinatorics,\u0000invariant theory, and factorization theory, etc. Apart from a few cases of\u0000finite groups, the precise value of the Davenport constant is unknown. Even in\u0000the abelian case, little is known beyond groups of rank at most two. On the\u0000other hand, for a fixed positive integer $r$, structural results characterizing\u0000which groups $G$ satisfy $mathsf D (G) = r$ are rare. We only know that there\u0000are finitely many such groups. In this paper, we study the classification of\u0000finite groups based on the Davenport constant.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206475","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}
引用次数: 0
3-manifold spine cyclic presentations with seldom seen Whitehead graphs 具有罕见怀特海图形的 3-manifold脊柱循环展示
arXiv - MATH - Group Theory Pub Date : 2024-08-30 DOI: arxiv-2408.17125
Gerald Williams
{"title":"3-manifold spine cyclic presentations with seldom seen Whitehead graphs","authors":"Gerald Williams","doi":"arxiv-2408.17125","DOIUrl":"https://doi.org/arxiv-2408.17125","url":null,"abstract":"We consider two families of cyclic presentations and show that, subject to\u0000certain conditions on the defining parameters, they are spines of closed\u00003-manifolds. For the first family, the Whitehead graphs have not previously\u0000been observed in this context, and the corresponding manifolds are lens spaces.\u0000The second family provides new examples where the reduced Whitehead graphs are\u0000those of the Fractional Fibonacci presentations; here the corresponding\u0000manifolds are often (but not always) hyperbolic.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206479","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}
引用次数: 0
Harmonious sequences in groups with a unique involution 具有独特内卷的群体中的和谐序列
arXiv - MATH - Group Theory Pub Date : 2024-08-29 DOI: arxiv-2408.16207
Mohammad Javaheri, Lydia de Wolf
{"title":"Harmonious sequences in groups with a unique involution","authors":"Mohammad Javaheri, Lydia de Wolf","doi":"arxiv-2408.16207","DOIUrl":"https://doi.org/arxiv-2408.16207","url":null,"abstract":"We study several combinatorial properties of finite groups that are related\u0000to the notions of sequenceability, R-sequenceability, and harmonious sequences.\u0000In particular, we show that in every abelian group $G$ with a unique involution\u0000$imath_G$ there exists a permutation $g_0,ldots, g_{m}$ of elements of $G\u0000backslash {imath_G}$ such that the consecutive sums $g_0+g_1,\u0000g_1+g_2,ldots, g_{m}+g_0$ also form a permutation of elements of $Gbackslash\u0000{imath_G}$. We also show that in every abelian group of order at least 4\u0000there exists a sequence containing each non-identity element of $G$ exactly\u0000twice such that the consecutive sums also contain each non-identity element of\u0000$G$ twice. We apply several results to the existence of transversals in Latin\u0000squares.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206481","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}
引用次数: 0
On neighborhoods in the enhanced power graph associated with a finite group 论与有限群相关的增强幂图中的邻域
arXiv - MATH - Group Theory Pub Date : 2024-08-29 DOI: arxiv-2408.16545
Mark L. Lewis, Carmine Monetta
{"title":"On neighborhoods in the enhanced power graph associated with a finite group","authors":"Mark L. Lewis, Carmine Monetta","doi":"arxiv-2408.16545","DOIUrl":"https://doi.org/arxiv-2408.16545","url":null,"abstract":"This article investigates neighborhoods' sizes in the enhanced power graph\u0000(as known as the cyclic graph) associated with a finite group. In particular,\u0000we characterize finite $p$-groups with the smallest maximum size for\u0000neighborhoods of nontrivial element in its enhanced power graph.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206480","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}
引用次数: 0
Associating hypergraphs defined on loops 循环上定义的关联超图
arXiv - MATH - Group Theory Pub Date : 2024-08-29 DOI: arxiv-2408.16459
Siddharth Malviy, Vipul Kakkar
{"title":"Associating hypergraphs defined on loops","authors":"Siddharth Malviy, Vipul Kakkar","doi":"arxiv-2408.16459","DOIUrl":"https://doi.org/arxiv-2408.16459","url":null,"abstract":"In this paper, we define a new hypergraph $mathcal{H(V,E)}$ on a loop $L$,\u0000where $mathcal{V}$ is the set of points of the loop $L$ and $mathcal{E}$ is\u0000the set of hyperedges $e={x,y,z}$ such that $x,y$ and $z$ associate in the\u0000order they are written. We call this hypergraph as the associating hypergraph\u0000on a loop $L$. We study certain properites of associating hypergraphs on the\u0000Moufang loop $M(D_n,2)$, where $D_n$ denotes the dihedral group of order $2n$.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206482","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}
引用次数: 0
Chirality and non-real elements in $G_2(q)$ 手性与$G_2(q)$中的非实数元素
arXiv - MATH - Group Theory Pub Date : 2024-08-28 DOI: arxiv-2408.15546
Sushil Bhunia, Amit Kulshrestha, Anupam Singh
{"title":"Chirality and non-real elements in $G_2(q)$","authors":"Sushil Bhunia, Amit Kulshrestha, Anupam Singh","doi":"arxiv-2408.15546","DOIUrl":"https://doi.org/arxiv-2408.15546","url":null,"abstract":"In this article, we determine the non-real elements--the ones that are not\u0000conjugate to their inverses--in the group $G = G_2(q)$ when $char(F_q)neq\u00002,3$. We use this to show that this group is chiral; that is, there is a word w\u0000such that $w(G)neq w(G)^{-1}$. We also show that most classical finite simple\u0000groups are achiral","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206488","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}
引用次数: 0
Weyl-invariants of totally disconnected locally compact groups acting cocompactly on buildings 完全断开局部紧密群作用于建筑物的韦尔不变式
arXiv - MATH - Group Theory Pub Date : 2024-08-28 DOI: arxiv-2408.15716
Ilaria Castellano, Bianca Marchionna, Thomas Weigel
{"title":"Weyl-invariants of totally disconnected locally compact groups acting cocompactly on buildings","authors":"Ilaria Castellano, Bianca Marchionna, Thomas Weigel","doi":"arxiv-2408.15716","DOIUrl":"https://doi.org/arxiv-2408.15716","url":null,"abstract":"In several instances, the invariants of compactly generated totally\u0000disconnected locally compact groups acting on locally finite buildings can be\u0000conveniently described via invariants of the Coxeter group representing the\u0000type of the building. For certain totally disconnected locally compact groups\u0000acting on buildings, we establish and collect several results concerning, for\u0000example, the rational discrete cohomological dimension (cf. Thm. A), the\u0000flat-rank (cf. Thm. C) and the number of ends (cf. Cor. K). Moreover, for an\u0000arbitrary compactly generated totally disconnected locally compact group, we\u0000express the number of ends in terms of its cohomology groups (cf. Thm. J).\u0000Furthermore, generalising a result of F. Haglund and F. Paulin, we prove that\u0000visual graph of groups decompositions of a Coxeter group $(W,S)$ can be used to\u0000construct trees from buildings of type $(W,S)$. We exploit the latter result to\u0000show that all $sigma$-compact totally disconnected locally compact groups\u0000acting chamber-transitively on buildings can be decomposed accordingly to any\u0000visual graph of groups decomposition of the type $(W,S)$ (cf. Thm. F and Cor.\u0000G).","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206485","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}
引用次数: 0
Uniform rank metric stability of Lie algebras, Lie groups and lattices 李代数、李群和网格的均匀秩度量稳定性
arXiv - MATH - Group Theory Pub Date : 2024-08-28 DOI: arxiv-2408.15614
Benjamin Bachner
{"title":"Uniform rank metric stability of Lie algebras, Lie groups and lattices","authors":"Benjamin Bachner","doi":"arxiv-2408.15614","DOIUrl":"https://doi.org/arxiv-2408.15614","url":null,"abstract":"We study uniform stability of discrete groups, Lie groups and Lie algebras in\u0000the rank metric, and the connections between uniform stability of these\u0000objects. We prove that semisimple Lie algebras are far from being flexibly\u0000$mathbb{C}$-stable, and that semisimple Lie groups and lattices in semisimple\u0000Lie groups of higher rank are not strictly $mathbb{C}$-stable. Furthermore, we\u0000prove that free groups are not uniformly flexibly $F$-stable over any field\u0000$F$.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206489","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}
引用次数: 0
Orbits of permutation groups with no derangements 无偏差排列群的轨道
arXiv - MATH - Group Theory Pub Date : 2024-08-28 DOI: arxiv-2408.16064
David Ellis, Scott Harper
{"title":"Orbits of permutation groups with no derangements","authors":"David Ellis, Scott Harper","doi":"arxiv-2408.16064","DOIUrl":"https://doi.org/arxiv-2408.16064","url":null,"abstract":"Let $G$ be a nontrivial finite permutation group of degree $n$. If $G$ is\u0000transitive, then a theorem of Jordan states that $G$ has a derangement.\u0000Equivalently, a finite group is never the union of conjugates of a proper\u0000subgroup. If $G$ is intransitive, then $G$ may fail to have a derangement, and\u0000this can happen even if $G$ has only two orbits, both of which have size\u0000$(1/2+o(1))n$. However, we conjecture that if $G$ has two orbits of size\u0000exactly $n/2$ then $G$ does have a derangement, and we prove this conjecture\u0000when $G$ acts primitively on at least one of the orbits. Equivalently, we\u0000conjecture that a finite group is never the union of conjugates of two proper\u0000subgroups of the same order, and we prove this conjecture when at least one of\u0000the subgroups is maximal. We prove other cases of the conjecture, and we\u0000highlight connections our results have with intersecting families of\u0000permutations and roots of polynomials modulo primes. Along the way, we also\u0000prove a linear variant on Isbell's conjecture regarding derangements of\u0000prime-power order.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206483","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}
引用次数: 0
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