Pub Date : 2024-10-21DOI: 10.1016/j.jalgebra.2024.10.011
Charles W. Eaton
We classify up to Morita equivalence all blocks whose defect groups are Suzuki 2-groups. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field, and in fact holds up to basic Morita equivalence. As a consequence Donovan's conjecture holds for Suzuki 2-groups. A corollary of the proof is that Suzuki Sylow 2-subgroups of finite groups with no nontrivial odd order normal subgroup are trivial intersection.
{"title":"Blocks whose defect groups are Suzuki 2-groups","authors":"Charles W. Eaton","doi":"10.1016/j.jalgebra.2024.10.011","DOIUrl":"10.1016/j.jalgebra.2024.10.011","url":null,"abstract":"<div><div>We classify up to Morita equivalence all blocks whose defect groups are Suzuki 2-groups. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field, and in fact holds up to basic Morita equivalence. As a consequence Donovan's conjecture holds for Suzuki 2-groups. A corollary of the proof is that Suzuki Sylow 2-subgroups of finite groups with no nontrivial odd order normal subgroup are trivial intersection.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142529843","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-21DOI: 10.1016/j.jalgebra.2024.10.018
Plamen Koshlukov , Felipe Yukihide Yasumura
We investigate the group gradings on the algebras of upper triangular matrices over an arbitrary field, viewed as Lie algebras. Classification results were obtained in 2017 by the same authors when the base field has characteristic different from 2. In this paper we provide streamlined proofs of these results. Moreover we present a complete classification of isomorphism classes of the group gradings on these algebras over an arbitrary field. Recall that two graded Lie algebras and are practically-isomorphic if there exists an (ungraded) algebra isomorphism that induces a graded-algebra isomorphism . We provide a classification of the practically-isomorphism classes of the group gradings on the Lie algebra of upper triangular matrices. The latter classification is a better alternative way to consider these gradings up to being essentially the same object. Finally, we investigate in details the case where the characteristic of the base field is 2, a topic that was neglected in previous works.
{"title":"Gradings on the algebra of triangular matrices as a Lie algebra: Revisited","authors":"Plamen Koshlukov , Felipe Yukihide Yasumura","doi":"10.1016/j.jalgebra.2024.10.018","DOIUrl":"10.1016/j.jalgebra.2024.10.018","url":null,"abstract":"<div><div>We investigate the group gradings on the algebras of upper triangular matrices over an arbitrary field, viewed as Lie algebras. Classification results were obtained in 2017 by the same authors when the base field has characteristic different from 2. In this paper we provide streamlined proofs of these results. Moreover we present a complete classification of isomorphism classes of the group gradings on these algebras over an arbitrary field. Recall that two graded Lie algebras <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> are practically-isomorphic if there exists an (ungraded) algebra isomorphism <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>→</mo><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> that induces a graded-algebra isomorphism <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>/</mo><mi>z</mi><mo>(</mo><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>→</mo><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>/</mo><mi>z</mi><mo>(</mo><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>. We provide a classification of the practically-isomorphism classes of the group gradings on the Lie algebra of upper triangular matrices. The latter classification is a better alternative way to consider these gradings up to being essentially the same object. Finally, we investigate in details the case where the characteristic of the base field is 2, a topic that was neglected in previous works.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554704","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-21DOI: 10.1016/j.jalgebra.2024.10.012
Marston D.E. Conder, Darius W. Young
This paper helps explain the prevalence of soluble groups among the automorphism groups of regular maps (at least for ‘small’ genus), by showing that every non-perfect hyperbolic ordinary triangle group has a smooth finite soluble quotient of derived length c for some , and infinitely many such quotients of derived length d for every .
{"title":"Soluble quotients of triangle groups","authors":"Marston D.E. Conder, Darius W. Young","doi":"10.1016/j.jalgebra.2024.10.012","DOIUrl":"10.1016/j.jalgebra.2024.10.012","url":null,"abstract":"<div><div>This paper helps explain the prevalence of soluble groups among the automorphism groups of regular maps (at least for ‘small’ genus), by showing that every non-perfect hyperbolic ordinary triangle group <span><math><msup><mrow><mi>Δ</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>r</mi><mo>)</mo><mo>=</mo><mo>〈</mo><mspace></mspace><mi>x</mi><mo>,</mo><mi>y</mi><mspace></mspace><mo>|</mo><mspace></mspace><msup><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>=</mo><msup><mrow><mi>y</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>=</mo><msup><mrow><mo>(</mo><mi>x</mi><mi>y</mi><mo>)</mo></mrow><mrow><mi>r</mi></mrow></msup><mo>=</mo><mn>1</mn><mspace></mspace><mo>〉</mo></math></span> has a smooth finite soluble quotient of derived length <em>c</em> for some <span><math><mi>c</mi><mo>≤</mo><mn>3</mn></math></span>, and infinitely many such quotients of derived length <em>d</em> for every <span><math><mi>d</mi><mo>></mo><mi>c</mi></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142560804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-21DOI: 10.1016/j.jalgebra.2024.10.015
Adrian Langer
We prove some strong results on approximation of strongly semistable bundles with vanishing numerical Chern classes by filtrations, whose quotients are line bundles of similar slope. This generalizes some earlier results of Parameswaran–Subramanian in the curve case and Koley–Parameswaran in the surface case and it confirms the conjecture posed by Koley and Parameswaran.
{"title":"Approximation of semistable bundles on smooth algebraic varieties","authors":"Adrian Langer","doi":"10.1016/j.jalgebra.2024.10.015","DOIUrl":"10.1016/j.jalgebra.2024.10.015","url":null,"abstract":"<div><div>We prove some strong results on approximation of strongly semistable bundles with vanishing numerical Chern classes by filtrations, whose quotients are line bundles of similar slope. This generalizes some earlier results of Parameswaran–Subramanian in the curve case and Koley–Parameswaran in the surface case and it confirms the conjecture posed by Koley and Parameswaran.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554701","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-21DOI: 10.1016/j.jalgebra.2024.09.036
Ioannis Papavasileiou , Dionysios Syrigos
Let G be a free product and the outer automorphism group of G. In this article using the theory of laminations we give a criterion for a subgroup H of to contain a nonabelian free subgroup. We also study the centraliser of a fully irreducible element of and the stabiliser of its associated lamination.
让 G 是自由积，Out(G) 是 G 的外自变群。在本文中，我们利用层积理论给出了 Out(G) 的子群 H 包含一个非阿贝尔自由子群的标准。我们还研究了 Out(G) 的完全不可约元素的中心化和其相关层叠的稳定化。
{"title":"Dynamics of fully irreducible automorphisms of free products","authors":"Ioannis Papavasileiou , Dionysios Syrigos","doi":"10.1016/j.jalgebra.2024.09.036","DOIUrl":"10.1016/j.jalgebra.2024.09.036","url":null,"abstract":"<div><div>Let <em>G</em> be a free product and <span><math><mrow><mi>Out</mi></mrow><mo>(</mo><mi>G</mi><mo>)</mo></math></span> the outer automorphism group of <em>G</em>. In this article using the theory of laminations we give a criterion for a subgroup <em>H</em> of <span><math><mrow><mi>Out</mi></mrow><mo>(</mo><mi>G</mi><mo>)</mo></math></span> to contain a nonabelian free subgroup. We also study the centraliser of a fully irreducible element of <span><math><mrow><mi>Out</mi></mrow><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and the stabiliser of its associated lamination.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142560805","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-21DOI: 10.1016/j.jalgebra.2024.10.014
Pham Hong Nam
In this paper, we define an unmixed torsion associated with a certain cohomological degree. We establish a connection between the Hilbert coefficients and the unmixed torsions of the module with respect to a parameter ideal generated by a d-sequence. As an application, we characterize the vanishing of the Hilbert coefficients through the depth of the module.
在本文中，我们定义了与特定同调度相关的非混合扭转。我们在模块的希尔伯特系数和非混合扭转之间建立了一种联系，这种联系是相对于由 d 序列生成的参数理想而言的。作为应用，我们通过模块的深度来描述希尔伯特系数的消失。
{"title":"Unmixed torsions and Hilbert coefficients of d-sequences","authors":"Pham Hong Nam","doi":"10.1016/j.jalgebra.2024.10.014","DOIUrl":"10.1016/j.jalgebra.2024.10.014","url":null,"abstract":"<div><div>In this paper, we define an unmixed torsion associated with a certain cohomological degree. We establish a connection between the Hilbert coefficients and the unmixed torsions of the module with respect to a parameter ideal generated by a <em>d</em>-sequence. As an application, we characterize the vanishing of the Hilbert coefficients through the depth of the module.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554797","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-16DOI: 10.1016/j.jalgebra.2024.09.029
Michael Turner
In our first paper, we looked at 2-generated primitive axial algebras of Monster type with skew axet . We continue our work by focusing on larger skew axets and classifying all such algebras with skew axets. This brings us one step closer to a complete classification of all 2-generated primitive axial algebras of Monster type.
{"title":"Skew axial algebras of Monster type II","authors":"Michael Turner","doi":"10.1016/j.jalgebra.2024.09.029","DOIUrl":"10.1016/j.jalgebra.2024.09.029","url":null,"abstract":"<div><div>In our first paper, we looked at 2-generated primitive axial algebras of Monster type with skew axet <span><math><msup><mrow><mi>X</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mn>1</mn><mo>+</mo><mn>2</mn><mo>)</mo></math></span>. We continue our work by focusing on larger skew axets and classifying all such algebras with skew axets. This brings us one step closer to a complete classification of all 2-generated primitive axial algebras of Monster type.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142529894","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-16DOI: 10.1016/j.jalgebra.2024.10.009
Truong Cong Quynh , M. Tamer Koşan , Jan Žemlička
The main aim of the paper is to describe the structure of modules and corresponding rings satisfying the property (P), which says that is embeddable into for each endomorphism α and which generalizes the morphic property. In particular, it is proved that the class of rings with the property (P) is closed under taking products and summands and contains unit regular rings. We also explain connections between the virtually internal cancellation property and the property (P) and characterize the structure of particular classes of rings satisfying the property (P).
{"title":"A new member of Nicholson's morphic folks","authors":"Truong Cong Quynh , M. Tamer Koşan , Jan Žemlička","doi":"10.1016/j.jalgebra.2024.10.009","DOIUrl":"10.1016/j.jalgebra.2024.10.009","url":null,"abstract":"<div><div>The main aim of the paper is to describe the structure of modules and corresponding rings satisfying the property (<em>P</em>), which says that <span><math><mi>M</mi><mo>/</mo><mi>im</mi><mo>(</mo><mi>α</mi><mo>)</mo></math></span> is embeddable into <span><math><mi>ker</mi><mo></mo><mo>(</mo><mi>α</mi><mo>)</mo></math></span> for each endomorphism <em>α</em> and which generalizes the morphic property. In particular, it is proved that the class of rings with the property (<em>P</em>) is closed under taking products and summands and contains unit regular rings. We also explain connections between the virtually internal cancellation property and the property (<em>P</em>) and characterize the structure of particular classes of rings satisfying the property (<em>P</em>).</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142529837","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-16DOI: 10.1016/j.jalgebra.2024.10.008
Raimundo Bastos , Irene N. Nakaoka , Noraí R. Rocco
In this paper we provide sufficient conditions for the non-abelian tensor product to be polycyclic/polycyclic-by-finite in terms of involved groups and derivative subgroups (cf. Theorem 1.1); we also give sufficient conditions for the (local) finiteness of the non-abelian tensor product of groups (cf. Theorem 1.2, Theorem 1.5). Furthermore, we deduce similar results for some related constructions associated to the non-abelian tensor products, such as the Schur multiplier of a pair of groups , the non-abelian q-tensor product , and homotopy pushout (cf. Section 5).
{"title":"On the finiteness of the non-abelian tensor product of groups","authors":"Raimundo Bastos , Irene N. Nakaoka , Noraí R. Rocco","doi":"10.1016/j.jalgebra.2024.10.008","DOIUrl":"10.1016/j.jalgebra.2024.10.008","url":null,"abstract":"<div><div>In this paper we provide sufficient conditions for the non-abelian tensor product <span><math><mi>G</mi><mo>⊗</mo><mi>H</mi></math></span> to be polycyclic/polycyclic-by-finite in terms of involved groups and derivative subgroups (cf. <span><span>Theorem 1.1</span></span>); we also give sufficient conditions for the (local) finiteness of the non-abelian tensor product of groups (cf. <span><span>Theorem 1.2</span></span>, <span><span>Theorem 1.5</span></span>). Furthermore, we deduce similar results for some related constructions associated to the non-abelian tensor products, such as the Schur multiplier of a pair of groups <span><math><mi>M</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>N</mi><mo>)</mo></math></span>, the non-abelian <em>q</em>-tensor product <span><math><mi>M</mi><msup><mrow><mo>⊗</mo></mrow><mrow><mi>q</mi></mrow></msup><mi>N</mi></math></span>, and homotopy pushout (cf. Section <span><span>5</span></span>).</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142529836","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-16DOI: 10.1016/j.jalgebra.2024.10.010
Lorenzo Stefanello , Cindy Tsang Sin Yi
Let be any finite Galois extension with Galois group G. It is known by Chase and Sweedler that the Hopf–Galois correspondence is injective for every Hopf–Galois structure on , but it need not be bijective in general. Hopf–Galois structures are known to be related to skew braces, and recently, the first-named author and Trappeniers proposed a new version of this connection with the property that the intermediate fields of in the image of the Hopf–Galois correspondence are in bijection with the left ideals of the associated skew brace. As an application, they classified the groups G for which the Hopf–Galois correspondence is bijective for every Hopf–Galois structure on any G-Galois extension. In this paper, using a similar approach, we shall classify the groups N for which the Hopf–Galois correspondence is bijective for every Hopf–Galois structure of type N on any Galois extension.
让 L/K 是任何具有伽罗瓦群 G 的有限伽罗瓦扩展。Chase 和 Sweedler 已知，对于 L/K 上的每一个 Hopf-Galois 结构，Hopf-Galois 对应都是注入式的，但在一般情况下不一定是双射的。众所周知，Hopf-Galois 结构与斜撑相关，最近，第一作者和 Trappeniers 提出了这一联系的新版本，其性质是在 Hopf-Galois 对应的映像中，L/K 的中间域与相关斜撑的左理想是双射的。作为一种应用，他们对任何 G-Galois 扩展上的每个 Hopf-Galois 结构的 Hopf-Galois 对应都是双射的群 G 进行了分类。在本文中，我们将采用类似的方法，对 N 群进行分类，对于这些群，在任何伽罗瓦扩展上的每一个 N 型 Hopf-Galois 结构，Hopf-Galois 对应都是双射的。
{"title":"Classification of the types for which every Hopf–Galois correspondence is bijective","authors":"Lorenzo Stefanello , Cindy Tsang Sin Yi","doi":"10.1016/j.jalgebra.2024.10.010","DOIUrl":"10.1016/j.jalgebra.2024.10.010","url":null,"abstract":"<div><div>Let <span><math><mi>L</mi><mo>/</mo><mi>K</mi></math></span> be any finite Galois extension with Galois group <em>G</em>. It is known by Chase and Sweedler that the Hopf–Galois correspondence is injective for every Hopf–Galois structure on <span><math><mi>L</mi><mo>/</mo><mi>K</mi></math></span>, but it need not be bijective in general. Hopf–Galois structures are known to be related to skew braces, and recently, the first-named author and Trappeniers proposed a new version of this connection with the property that the intermediate fields of <span><math><mi>L</mi><mo>/</mo><mi>K</mi></math></span> in the image of the Hopf–Galois correspondence are in bijection with the left ideals of the associated skew brace. As an application, they classified the groups <em>G</em> for which the Hopf–Galois correspondence is bijective for every Hopf–Galois structure on any <em>G</em>-Galois extension. In this paper, using a similar approach, we shall classify the groups <em>N</em> for which the Hopf–Galois correspondence is bijective for every Hopf–Galois structure of type <em>N</em> on any Galois extension.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142529844","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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