{"title":"On finite groups whose power graph is claw-free","authors":"Pallabi Manna, Santanu Mandal, Andrea Lucchini","doi":"arxiv-2407.20110","DOIUrl":"https://doi.org/arxiv-2407.20110","url":null,"abstract":"A graph is called claw-free if it contains no induced subgraph isomorphic to\u0000the complete bipartite graph $K_{1, 3}$. The undirected power graph of a group\u0000$G$ has vertices the elements of $G$, with an edge between $g_1$ and $g_2$ if\u0000one of the two cyclic subgroups $langle g_1rangle, langle g_2rangle$ is\u0000contained in the other. It is denoted by $P(G)$. The reduced power graph,\u0000denoted by $P^*(G),$ is the subgraph of $P(G)$ induced by the non-identity\u0000elements. The main purpose of this paper is to explore the finite groups whose\u0000reduced power graph is claw-free. In particular we prove that if $P^*(G)$ is\u0000claw-free, then either $G$ is solvable or $G$ is an almost simple group. In the\u0000second case the socle of $G$ is isomorphic to $PSL(2,q)$ for suitable choices\u0000of $q$. Finally we prove that if $P^*(G)$ is claw-free, then the order of $G$\u0000is divisible by at most 5 different primes.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"168 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141862900","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}
{"title":"$GL_n(mathbb{F}_q)$-analogues of some properties of $n$-cycles in $mathfrak{S}_n$","authors":"Joel Brewster Lewis","doi":"arxiv-2407.20347","DOIUrl":"https://doi.org/arxiv-2407.20347","url":null,"abstract":"We give analogues in the finite general linear group of two elementary\u0000results concerning long cycles and transpositions in the symmetric group:\u0000first, that the long cycles are precisely the elements whose minimum-length\u0000factorizations into transpositions yield a generating set, and second, that a\u0000long cycle together with an appropriate transposition generates the whole\u0000symmetric group.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141862898","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}
{"title":"Automorphism group of the graph $A(n,k,r)$","authors":"Junyao Pan","doi":"arxiv-2407.19745","DOIUrl":"https://doi.org/arxiv-2407.19745","url":null,"abstract":"Let $[n]^{(k)}$ be the set of all ordered $k$-tuples of distinct elements in\u0000$[n]={1,2,...,n}$. The $(n,k,r)$-arrangement graph $A(n,k,r)$ with $1leq\u0000rleq kleq n$, is the graph with vertex set $[n]^{(k)}$ and with two\u0000$k$-tuples are adjacent if they differ in exactly $r$ coordinates. In this\u0000manuscript, we characterize the full automorphism groups of $A(n,k,r)$ in the\u0000cases that $1leq r=kleq n$ and $r=2<k=n$. Thus, we resolve two special cases\u0000of an open problem proposed by Fu-Gang Yin, Yan-Quan Feng, Jin-Xin Zhou and\u0000Yu-Hong Guo. In addition, we conclude with a bold conjecture.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"99 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141862901","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}
{"title":"Essential subgroups and essential extensions","authors":"Sourav Koner, Biswajit Mitra","doi":"arxiv-2407.19309","DOIUrl":"https://doi.org/arxiv-2407.19309","url":null,"abstract":"The notion of essential submodules and essential extensions of modules are\u0000extended to groups (typically nonabelian), and several necessary and sufficient\u0000conditions for a group to possess a proper essential subgroup are investigated.\u0000Further, we have completely characterized groups that do not possess a proper\u0000essential extension. These observations are used in concluding several\u0000properties of groups having essential subgroups. Finally, a short proof of the\u0000well-known theorem of Eilenberg and Moore that the only injective object in the\u0000category of groups is the trivial group is given.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141862902","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}
{"title":"Automorphism Group of the Holomorph of a Cyclic Group","authors":"Kazuki Sato","doi":"arxiv-2407.18435","DOIUrl":"https://doi.org/arxiv-2407.18435","url":null,"abstract":"We show that the holomorph of a cyclic group of order $n$ is isomorphic to\u0000its own automophism group when $n$ is twice of a power of an odd prime.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"145 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141862906","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}
{"title":"Benjamini-Schramm vs Plancherel convergence","authors":"Giacomo Gavelli, Claudius Kamp","doi":"arxiv-2407.18822","DOIUrl":"https://doi.org/arxiv-2407.18822","url":null,"abstract":"We show that Plancherel convergence is strictly stronger than\u0000Benjamini-Schramm convergence. In order to do so, we introduce two criteria for\u0000Plancherel and Benjamini-Schramm convergence in terms of counting functions of\u0000the length spectrum.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"110 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863033","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}
{"title":"Automorphisms of the two-sided shift and the Higman--Thompson groups III: extensions","authors":"Feyishayo Olukoya","doi":"arxiv-2407.18720","DOIUrl":"https://doi.org/arxiv-2407.18720","url":null,"abstract":"We aim to interpret important constructions in the theory of automorphisms of\u0000the shift dynamical system in terms of subgroups $mathcal{L}_{n,r}$ of the\u0000outer-automorphism groups $mathcal{O}_{n,r}$ of the Higman--Thompson group\u0000$G_{n,r}$, and to extend results and techniques in\u0000$operatorname{Aut}(X_n^{mathbb{Z}}, sigma_{n})$ to the groups of\u0000automorphisms $operatorname{Aut}(G_{n,r})$ and outerautomrphisms of the\u0000Higman--Thompson group $G_{n,r}$. Our mains results are a concrete realisation of the \"inert subgroup\",\u0000important subgroup in the study of automorphism groups of shift spaces, as a\u0000subgroup $mathcal{K}_{n}$ of $mathcal{L}_{n,n-1}$. Using this realisation, we show that the $operatorname{Aut}(G_{n,r})$\u0000contains an isomorphic copy of $operatorname{Aut}(X_{m}^{mathbb{Z}},\u0000sigma_{m})$ for all $m ge 2$. A survey of the literature then yields that $operatorname{Aut}(G_{n,r})$\u0000contains isomorphic copies of finite groups, finitely generated abelian groups,\u0000free groups, free products of finite groups, fundamental groups of 2-manifolds,\u0000graph groups and countable locally finite residually finite groups to name a\u0000few. We extend a result for $operatorname{Aut}(X_n^{mathbb{Z}}, sigma_{n})$ to\u0000the group $mathcal{O}_{n,n-1}$. The homeomorphism\u0000$overleftarrow{phantom{a}}$ of $X_n^{mathbb{Z}}$ which maps a sequence\u0000$(x_i)_{i in mathbb{Z}}$ to the sequence $(y_{i})_{i in mathbb{Z}}$ defined\u0000such that $y_{i} = x_{-i}$ induces an automorphism\u0000$overleftarrow{mathfrak{r}}$ of $operatorname{Aut}(X_n^{mathbb{Z}},\u0000sigma_{n})$, and consequently, an automorphism of $mathcal{L}_{n}$. We extend\u0000the automorphism $overleftarrow{{mathfrak{r}}}$ to the group\u0000$mathcal{O}_{n,n-1}$. In a forthcoming article, we demonstrate that the group $mathcal{O}_{n}$ is\u0000isomorphic to the mapping class group of the full two-sided shift over $n$\u0000letters.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"170 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141862904","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}
{"title":"Nearly-linear solution to the word problem for 3-manifold groups","authors":"Alessandro Sisto, Stefanie Zbinden","doi":"arxiv-2407.18029","DOIUrl":"https://doi.org/arxiv-2407.18029","url":null,"abstract":"We show that the word problem for any 3-manifold group is solvable in time\u0000$O(nlog^3 n)$. Our main contribution is the proof that the word problem for\u0000admissible graphs of groups, in the sense of Croke and Kleiner, is solvable in\u0000$O(nlog n)$; this covers fundamental groups of non-geometric graph manifolds.\u0000Similar methods also give that the word problem for free products can be solved\u0000\"almost as quickly\" as the word problem in the factors.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141785611","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}
{"title":"Are free groups of different ranks bi-invariantly quasi-isometric?","authors":"Jarek Kędra, Assaf Libman","doi":"arxiv-2407.18027","DOIUrl":"https://doi.org/arxiv-2407.18027","url":null,"abstract":"We prove that a homomorphism between free groups of finite rank equipped with\u0000the bi-invariant word metrics is a quasi-isometry if and only if it is an\u0000isomorphism.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141776221","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}
{"title":"Koszul Resolutions over Free Incomplete Tambara Functors for Cyclic $p$-Groups","authors":"David Mehrle, J. D. Quigley, Michael Stahlhauer","doi":"arxiv-2407.18382","DOIUrl":"https://doi.org/arxiv-2407.18382","url":null,"abstract":"In equivariant algebra, Mackey functors replace abelian groups and incomplete\u0000Tambara functors replace commutative rings. In this context, we prove that\u0000equivariant Hochschild homology can sometimes be computed using Mackey\u0000functor-valued Tor. To compute these Tor Mackey functors for odd primes $p$, we\u0000define cyclic-$p$-group-equivariant analogues of the Koszul resolution which\u0000resolve the Burnside Mackey functor (the analogue of the integers) as a module\u0000over free incomplete Tambara functors (the analogue of polynomial rings). We\u0000apply these Koszul resolutions to compute Mackey functor-valued Hochschild\u0000homology of free incomplete Tambara functors for cyclic groups of odd prime\u0000order and for the cyclic group of order 9.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"54 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141862905","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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