The well-known theory of Pontryagin duality provides a strong connection between the homology and cohomology theories of a profinite group in appropriate categories. A construction for taking the `profinite direct sum' of an infinite family of profinite modules indexed over a profinite space has been found to be useful in the study of homology of profinite groups, but hitherto the appropriate dual construction for studying cohomology with coefficients in discrete modules has not been studied. This paper remedies this gap in the theory.
{"title":"Pontryagin duality and sheaves of profinite modules","authors":"Gareth Wilkes","doi":"arxiv-2408.13059","DOIUrl":"https://doi.org/arxiv-2408.13059","url":null,"abstract":"The well-known theory of Pontryagin duality provides a strong connection\u0000between the homology and cohomology theories of a profinite group in\u0000appropriate categories. A construction for taking the `profinite direct sum' of\u0000an infinite family of profinite modules indexed over a profinite space has been\u0000found to be useful in the study of homology of profinite groups, but hitherto\u0000the appropriate dual construction for studying cohomology with coefficients in\u0000discrete modules has not been studied. This paper remedies this gap in the\u0000theory.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206503","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}
In this paper, we introduce the concept of periodic tiling (PT) property for finite abelian groups. A group has the PT property if any non-periodic set that tiles the group by translation has a periodic tiling complement. This property extends the scope beyond groups with the Haj'os property. We classify all cyclic groups having the PT property. Additionally, we construct groups that possess the PT property but without the Haj'os property. As byproduct, we identify new groups for which the implication ``Tile $Longrightarrow$ Spectral" holds. For elementary $p$-groups having the PT property, we show that a tile must be a complete set of representatives of the cosets of some subgroup, by analyzing the structure of tiles.
{"title":"Periodicity of tiles in finite Abelian groups","authors":"Shilei Fan, Tao Zhang","doi":"arxiv-2408.12901","DOIUrl":"https://doi.org/arxiv-2408.12901","url":null,"abstract":"In this paper, we introduce the concept of periodic tiling (PT) property for\u0000finite abelian groups. A group has the PT property if any non-periodic set that\u0000tiles the group by translation has a periodic tiling complement. This property\u0000extends the scope beyond groups with the Haj'os property. We classify all\u0000cyclic groups having the PT property. Additionally, we construct groups that\u0000possess the PT property but without the Haj'os property. As byproduct, we\u0000identify new groups for which the implication ``Tile $Longrightarrow$\u0000Spectral\" holds. For elementary $p$-groups having the PT property, we show that\u0000a tile must be a complete set of representatives of the cosets of some\u0000subgroup, by analyzing the structure of tiles.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206505","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}
The solvable Baumslag Solitar groups $text{BS}(1,n)$ each admit a canonical model space, $X_n$. We give a complete classification of lattices in $G_n = text{Isom}^+(X_n)$ and find that such lattices fail to be strongly rigid$unicode{x2014}$there are automorphisms of lattices $Gamma subset G_n$ which do not extend to $G_n$$unicode{x2014}$but do satisfy a weaker form of rigidity: for all isomorphic lattices $Gamma_1,Gamma_2subset G_n$, there is an automorphism $rho in text{Aut}(G_n)$ so that $rho(Gamma_1) = Gamma_2$.
{"title":"Solvable Baumslag-Solitar Lattices","authors":"Noah Caplinger","doi":"arxiv-2408.13381","DOIUrl":"https://doi.org/arxiv-2408.13381","url":null,"abstract":"The solvable Baumslag Solitar groups $text{BS}(1,n)$ each admit a canonical\u0000model space, $X_n$. We give a complete classification of lattices in $G_n =\u0000text{Isom}^+(X_n)$ and find that such lattices fail to be strongly\u0000rigid$unicode{x2014}$there are automorphisms of lattices $Gamma subset G_n$\u0000which do not extend to $G_n$$unicode{x2014}$but do satisfy a weaker form of\u0000rigidity: for all isomorphic lattices $Gamma_1,Gamma_2subset G_n$, there is\u0000an automorphism $rho in text{Aut}(G_n)$ so that $rho(Gamma_1) = Gamma_2$.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206501","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}
A subset ${g_1, ldots , g_d}$ of a finite group $G$ invariably generates $G$ if ${g_1^{x_1}, ldots , g_d^{x_d}}$ generates $G$ for every choice of $x_i in G$. The Chebotarev invariant $C(G)$ of $G$ is the expected value of the random variable $n$ that is minimal subject to the requirement that $n$ randomly chosen elements of $G$ invariably generate $G$. In this paper, we show that if $G$ is a nonabelian finite simple group, then $C(G)$ is absolutely bounded. More generally, we show that if $G$ is a direct product of $k$ nonabelian finite simple groups, then $C(G)=log{k}/log{alpha(G)}+O(1)$, where $alpha$ is an invariant completely determined by the proportion of derangements of the primitive permutation actions of the factors in $G$. It follows from the proof of the Boston-Shalev conjecture that $C(G)=O(log{k})$. We also derive sharp bounds on the expected number of generators for $G$.
{"title":"The Chebotarev invariant for direct products of nonabelian finite simple groups","authors":"Jessica Anzanello, Andrea Lucchini, Gareth Tracey","doi":"arxiv-2408.12298","DOIUrl":"https://doi.org/arxiv-2408.12298","url":null,"abstract":"A subset ${g_1, ldots , g_d}$ of a finite group $G$ invariably generates\u0000$G$ if ${g_1^{x_1}, ldots , g_d^{x_d}}$ generates $G$ for every choice of\u0000$x_i in G$. The Chebotarev invariant $C(G)$ of $G$ is the expected value of\u0000the random variable $n$ that is minimal subject to the requirement that $n$\u0000randomly chosen elements of $G$ invariably generate $G$. In this paper, we show\u0000that if $G$ is a nonabelian finite simple group, then $C(G)$ is absolutely\u0000bounded. More generally, we show that if $G$ is a direct product of $k$\u0000nonabelian finite simple groups, then $C(G)=log{k}/log{alpha(G)}+O(1)$,\u0000where $alpha$ is an invariant completely determined by the proportion of\u0000derangements of the primitive permutation actions of the factors in $G$. It\u0000follows from the proof of the Boston-Shalev conjecture that $C(G)=O(log{k})$.\u0000We also derive sharp bounds on the expected number of generators for $G$.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206510","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}
Bruno Aaron Cisneros de la Cruz, María Cumplido, Islam Foniqi
We generalize the retractions to standard parabolic subgroups for even Artin groups to FC-type Artin groups and other more general families. We prove that these retractions uniquely extend to any parabolic subgroup. We use retractions to generalize the results of Antol'in and Foniqi that reduce the problem of intersection of parabolic subgroups to weaker conditions. As a corollary, we characterize coherence for the FC case.
我们将偶数阿尔丁群的标准抛物面子群的回缩推广到 FC 型阿尔丁群和其他更一般的群族。我们证明这些回缩唯一地扩展到任何抛物线子群。我们利用溯源推广了 Antol'in 和 Foniqi 的结果，这些结果将抛物线子群的交集问题简化为较弱的条件。作为推论，我们描述了 FC 情况下的一致性。
{"title":"On Artin groups admitting retractions to parabolic subgroups","authors":"Bruno Aaron Cisneros de la Cruz, María Cumplido, Islam Foniqi","doi":"arxiv-2408.12291","DOIUrl":"https://doi.org/arxiv-2408.12291","url":null,"abstract":"We generalize the retractions to standard parabolic subgroups for even Artin\u0000groups to FC-type Artin groups and other more general families. We prove that\u0000these retractions uniquely extend to any parabolic subgroup. We use retractions\u0000to generalize the results of Antol'in and Foniqi that reduce the problem of\u0000intersection of parabolic subgroups to weaker conditions. As a corollary, we\u0000characterize coherence for the FC case.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206506","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}
The Lannes-Quillen theorem relates the mod-$p$ cohomology of a finite group $G$ with the mod-$p$ cohomology of centralizers of abelian elementary $p$-subgroups of $G$, for $p>0$ a prime number. This theorem was extended to profinite groups whose mod-$p$ cohomology algebra is finitely generated by Henn. In a weaker form, the Lannes-Quillen theorem was then extended by Symonds to arbitrary profinite groups. Building on Symonds' result, we formulate and prove a full version of this theorem for all profinite groups. For this purpose, we develop a theory of products for families of discrete torsion modules, parameterized by a profinite space, which is dual, in a very precise sense, to the theory of coproducts for families of profinite modules, parameterized by a profinite space, developed by Haran, Melnikov and Ribes. In the last section, we give applications to the problem of conjugacy separability of $p$-torsion elements and finite $p$-subgroups.
{"title":"Lannes' $T$-functor and mod-$p$ cohomology of profinite groups","authors":"Marco Boggi","doi":"arxiv-2408.12488","DOIUrl":"https://doi.org/arxiv-2408.12488","url":null,"abstract":"The Lannes-Quillen theorem relates the mod-$p$ cohomology of a finite group\u0000$G$ with the mod-$p$ cohomology of centralizers of abelian elementary\u0000$p$-subgroups of $G$, for $p>0$ a prime number. This theorem was extended to\u0000profinite groups whose mod-$p$ cohomology algebra is finitely generated by\u0000Henn. In a weaker form, the Lannes-Quillen theorem was then extended by Symonds\u0000to arbitrary profinite groups. Building on Symonds' result, we formulate and\u0000prove a full version of this theorem for all profinite groups. For this\u0000purpose, we develop a theory of products for families of discrete torsion\u0000modules, parameterized by a profinite space, which is dual, in a very precise\u0000sense, to the theory of coproducts for families of profinite modules,\u0000parameterized by a profinite space, developed by Haran, Melnikov and Ribes. In\u0000the last section, we give applications to the problem of conjugacy separability\u0000of $p$-torsion elements and finite $p$-subgroups.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206504","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}
Satish Kumar, Harshdeep Singh, Indivar Gupta, Ashok Ji Gupta
In this paper, we propose a novel construction for a symmetric encryption scheme, referred as SEBQ which is based on the structure of quasigroup. We utilize concepts of chaining like mode of operation and present a block cipher with in-built properties. We prove that SEBQ shows resistance against chosen plaintext attack (CPA) and by applying unbalanced Feistel transformation [19], it achieves security against chosen ciphertext attacks (CCA). Subsequently, we conduct an assessment of the randomness of the proposed scheme by running the NIST test suite and we analyze the impact of the initial vector, secret key and plaintext on ciphertext through an avalanche effect analysis. We also compare the results with existing schemes based on quasigroups [11,46]. Moreover, we analyze the computational complexity in terms of number of operations needed for encryption and decryption process.
{"title":"Symmetric Encryption Scheme Based on Quasigroup Using Chained Mode of Operation","authors":"Satish Kumar, Harshdeep Singh, Indivar Gupta, Ashok Ji Gupta","doi":"arxiv-2408.04490","DOIUrl":"https://doi.org/arxiv-2408.04490","url":null,"abstract":"In this paper, we propose a novel construction for a symmetric encryption\u0000scheme, referred as SEBQ which is based on the structure of quasigroup. We\u0000utilize concepts of chaining like mode of operation and present a block cipher\u0000with in-built properties. We prove that SEBQ shows resistance against chosen\u0000plaintext attack (CPA) and by applying unbalanced Feistel transformation [19],\u0000it achieves security against chosen ciphertext attacks (CCA). Subsequently, we\u0000conduct an assessment of the randomness of the proposed scheme by running the\u0000NIST test suite and we analyze the impact of the initial vector, secret key and\u0000plaintext on ciphertext through an avalanche effect analysis. We also compare\u0000the results with existing schemes based on quasigroups [11,46]. Moreover, we\u0000analyze the computational complexity in terms of number of operations needed\u0000for encryption and decryption process.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934014","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}
This paper is concerned with rational curves on real classical groups. Our contributions are three-fold: (i) We determine the structure of quadratic rational curves on real classical groups. As a consequence, we completely classify quadratic rational curves on $mathrm{U}_n$, $mathrm{O}_n(mathbb{R})$, $mathrm{O}_{n-1,1}(mathbb{R})$ and $mathrm{O}_{n-2,2}(mathbb{R})$. (ii) We prove a decomposition theorem for rational curves on real classical groups, which can be regarded as a non-commutative generalization of the fundamental theorem of algebra and partial fraction decomposition. (iii) As an application of (i) and (ii), we generalize Kempe's Universality Theorem to rational curves on homogeneous spaces.
{"title":"Rational Curves on Real Classical Groups","authors":"Zijia Li, Ke Ye","doi":"arxiv-2408.04453","DOIUrl":"https://doi.org/arxiv-2408.04453","url":null,"abstract":"This paper is concerned with rational curves on real classical groups. Our\u0000contributions are three-fold: (i) We determine the structure of quadratic\u0000rational curves on real classical groups. As a consequence, we completely\u0000classify quadratic rational curves on $mathrm{U}_n$,\u0000$mathrm{O}_n(mathbb{R})$, $mathrm{O}_{n-1,1}(mathbb{R})$ and\u0000$mathrm{O}_{n-2,2}(mathbb{R})$. (ii) We prove a decomposition theorem for\u0000rational curves on real classical groups, which can be regarded as a\u0000non-commutative generalization of the fundamental theorem of algebra and\u0000partial fraction decomposition. (iii) As an application of (i) and (ii), we\u0000generalize Kempe's Universality Theorem to rational curves on homogeneous\u0000spaces.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934011","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}
Kuz'min (1996) characterized groups having Wirtinger presentations in relation to their second group homology. In this paper, we further refine the relation between these groups and their second group homology.
{"title":"Groups having Wirtinger presentations and the second group homology","authors":"Toshiyuki Akita, Sota Takase","doi":"arxiv-2408.04265","DOIUrl":"https://doi.org/arxiv-2408.04265","url":null,"abstract":"Kuz'min (1996) characterized groups having Wirtinger presentations in\u0000relation to their second group homology. In this paper, we further refine the\u0000relation between these groups and their second group homology.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934025","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}
We give a description of finitely generated prosoluble subgroups of the profinite completion of $3$-manifold groups and virtually compact special groups.
我们给出了 3$-manifold群的无限完备群和虚紧凑特殊群的有限生成的前可溶子群的描述。
{"title":"Prosoluble subgroups of the profinite completion of the fundamental group of compact 3-manifolds","authors":"Lucas C. Lopes, Pavel A. Zalesskii","doi":"arxiv-2408.04152","DOIUrl":"https://doi.org/arxiv-2408.04152","url":null,"abstract":"We give a description of finitely generated prosoluble subgroups of the\u0000profinite completion of $3$-manifold groups and virtually compact special\u0000groups.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"193 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934012","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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