Journal of Group Theory最新文献

{"title":"Finitely generated metabelian groups arising from integer polynomials","authors":"Derek J. S. Robinson","doi":"10.1515/jgth-2023-0046","DOIUrl":"https://doi.org/10.1515/jgth-2023-0046","url":null,"abstract":"Abstract It is shown that there is a finitely generated metabelian group of finite torsion-free rank associated with each non-constant integer polynomial. It is shown how many structural properties of the group can be detected by inspecting the polynomial.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135477400","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hall classes in linear groups","authors":"Francesco de Giovanni, Marco Trombetti, Bertram A. F. Wehrfritz","doi":"10.1515/jgth-2023-0063","DOIUrl":"https://doi.org/10.1515/jgth-2023-0063","url":null,"abstract":"Abstract A well-known theorem of Philip Hall states that if a group 𝐺 has a nilpotent normal subgroup 𝑁 such that <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:msup> <m:mi>N</m:mi> <m:mo>′</m:mo> </m:msup> </m:mrow> </m:math> G/N^{prime} is nilpotent, then 𝐺 itself is nilpotent. We say that a group class 𝔛 is a Hall class if it contains every group 𝐺 admitting a nilpotent normal subgroup 𝑁 such that <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:msup> <m:mi>N</m:mi> <m:mo>′</m:mo> </m:msup> </m:mrow> </m:math> G/N^{prime} belongs to 𝔛. Examples have been given in [F. de Giovanni, M. Trombetti and B. A. F. Wehfritz, Hall classes of groups, to appear] to show that finite-by-𝔛 groups do not form a Hall class for many natural choices of the Hall class 𝔛. Although these examples are often linear, our aim here is to prove that the situation is much better within certain natural subclasses of the universe of linear groups.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136129016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A closure operator on the subgroup lattice of GL(𝑛,𝑞) and PGL(𝑛,𝑞) in relation to the zeros of the Möbius function","authors":"Luca Di Gravina","doi":"10.1515/jgth-2023-0021","DOIUrl":"https://doi.org/10.1515/jgth-2023-0021","url":null,"abstract":"Abstract Let <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi mathvariant=\"double-struck\">F</m:mi> <m:mi>q</m:mi> </m:msub> </m:math> mathbb{F}_{q} be the finite field with 𝑞 elements and consider the 𝑛-dimensional <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi mathvariant=\"double-struck\">F</m:mi> <m:mi>q</m:mi> </m:msub> </m:math> mathbb{F}_{q} -vector space <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>V</m:mi> <m:mo>=</m:mo> <m:msubsup> <m:mi mathvariant=\"double-struck\">F</m:mi> <m:mi>q</m:mi> <m:mi>n</m:mi> </m:msubsup> </m:mrow> </m:math> V=mathbb{F}_{q}^{n} . In this paper, we define a closure operator on the subgroup lattice of the group <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>G</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mi>PGL</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>V</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> G=mathrm{PGL}(V) . Let 𝜇 denote the Möbius function of this lattice. The aim is to use this closure operator to characterize subgroups 𝐻 of 𝐺 for which <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>μ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>H</m:mi> <m:mo>,</m:mo> <m:mi>G</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>≠</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> mu(H,G)neq 0 . Moreover, we establish a polynomial bound on the number <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>c</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>m</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> c(m) of closed subgroups 𝐻 of index 𝑚 in 𝐺 for which the lattice of 𝐻-invariant subspaces of 𝑉 is isomorphic to a product of chains. This bound depends only on 𝑚 and not on the choice of 𝑛 and 𝑞. It is achieved by considering a similar closure operator for the subgroup lattice of <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>GL</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>V</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> mathrm{GL}(V) and the same results proven for this group.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135011557","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the strong connectivity of the 2-Engel graphs of almost simple groups","authors":"Francesca Dalla Volta, Fabio Mastrogiacomo, Pablo Spiga","doi":"10.1515/jgth-2023-0060","DOIUrl":"https://doi.org/10.1515/jgth-2023-0060","url":null,"abstract":"Abstract The Engel graph of a finite group 𝐺 is a directed graph encoding the pairs of elements in 𝐺 satisfying some Engel word. Recent work of Lucchini and the third author shows that, except for a few well-understood cases, the Engel graphs of almost simple groups are strongly connected. In this paper, we give a refinement to this analysis.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135011556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Relative stable equivalences of Morita type for the principal blocks of finite groups and relative Brauer indecomposability","authors":"Naoko Kunugi, Kyoichi Suzuki","doi":"10.1515/jgth-2023-0033","DOIUrl":"https://doi.org/10.1515/jgth-2023-0033","url":null,"abstract":"Abstract We discuss representations of finite groups having a common central 𝑝-subgroup 𝑍, where 𝑝 is a prime number. For the principal 𝑝-blocks, we give a method of constructing a relative 𝑍-stable equivalence of Morita type, which is a generalization of stable equivalence of Morita type and was introduced by Wang and Zhang in a more general setting. Then we generalize Linckelmann’s results on stable equivalences of Morita type to relative 𝑍-stable equivalences of Morita type. We also introduce the notion of relative Brauer indecomposability, which is a generalization of the notion of Brauer indecomposability. We give an equivalent condition for Scott modules to be relatively Brauer indecomposable, which is an analog of that given by Ishioka and the first author.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135010788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Graphical complexes of groups","authors":"Tomasz Prytuła","doi":"10.1515/jgth-2021-0118","DOIUrl":"https://doi.org/10.1515/jgth-2021-0118","url":null,"abstract":"Abstract We introduce graphical complexes of groups, which can be thought of as a generalisation of Coxeter systems with 1-dimensional nerves. We show that these complexes are strictly developable, and we equip the resulting Basic Construction with three structures of non-positive curvature: piecewise linear <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>CAT</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mn>0</m:mn> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> mathrm{CAT}(0) , <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>C</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mn>6</m:mn> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> C(6) graphical small cancellation, and a systolic one. We then use these structures to establish various properties of the fundamental groups of these complexes, such as biautomaticity and the Tits Alternative. We isolate an easily checkable condition implying hyperbolicity of the fundamental groups, and we construct some non-hyperbolic examples. We also briefly discuss a parallel theory of <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>C</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mn>4</m:mn> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> C(4) - <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>T</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mn>4</m:mn> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> T(4) graphical complexes of groups and outline their basic properties.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135488732","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Finite 𝑝-groups of class two with a small multiple holomorph","authors":"A. Caranti, Cindy (Sin Yi) Tsang","doi":"10.1515/jgth-2023-0054","DOIUrl":"https://doi.org/10.1515/jgth-2023-0054","url":null,"abstract":"Abstract We consider the quotient group T ⁢ ( G ) T(G) of the multiple holomorph by the holomorph of a finite 𝑝-group 𝐺 of class two for an odd prime 𝑝. By work of the first-named author, we know that T ⁢ ( G ) T(G) contains a cyclic subgroup of order p r − 1 ⁢ ( p − 1 ) p^{r-1}(p-1) , where p r p^{r} is the exponent of the quotient of 𝐺 by its center. In this paper, we shall exhibit examples of 𝐺 (with r = 1 r=1 ) such that T ⁢ ( G ) T(G) has order exactly p − 1 p-1 , which is as small as possible.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78735271","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Presentations of Schur covers of braid groups","authors":"Toshiyuki Akita, Rikako Kawasaki, T. Satoh","doi":"10.1515/jgth-2023-0014","DOIUrl":"https://doi.org/10.1515/jgth-2023-0014","url":null,"abstract":"Abstract In this paper, we consider several basic facts of Schur covers of the symmetric groups and braid groups. In particular, we give explicit presentations of Schur covers of braid groups.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73543964","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A characterization of finite groups having a single Galois conjugacy class of certain irreducible characters","authors":"Yuedi Zeng, Dongfang Yang","doi":"10.1515/jgth-2022-0215","DOIUrl":"https://doi.org/10.1515/jgth-2022-0215","url":null,"abstract":"Abstract Let 𝐺 be a finite group and let Irr s ⁢ ( G ) mathrm{Irr}_{mathfrak{s}}(G) be the set of irreducible complex characters 𝜒 of 𝐺 such that χ ⁢ ( 1 ) 2 chi(1)^{2} does not divide the index of the kernel of 𝜒. In this paper, we classify the finite groups 𝐺 for which any two characters in Irr s ⁢ ( G ) mathrm{Irr}_{mathfrak{s}}(G) are Galois conjugate. In particular, we show that such groups are solvable with Fitting height 2.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74441398","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A classification of the prime graphs of pseudo-solvable groups","authors":"Ziyu Huang, Thomas Michael Keller, Shane Kissinger, Wen Plotnick, Maya Roma, Yong Yang","doi":"10.1515/jgth-2023-0018","DOIUrl":"https://doi.org/10.1515/jgth-2023-0018","url":null,"abstract":"The prime graph <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"normal\">Γ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0018_ineq_0001.png\" /> <jats:tex-math>Gamma(G)</jats:tex-math> </jats:alternatives> </jats:inline-formula> of a finite group 𝐺 (also known as the Gruenberg–Kegel graph) has as its vertices the prime divisors of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">|</m:mo> <m:mi>G</m:mi> <m:mo stretchy=\"false\">|</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0018_ineq_0002.png\" /> <jats:tex-math>lvert Grvert</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:mtext>-</m:mtext> <m:mo>⁢</m:mo> <m:mi>q</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0018_ineq_0003.png\" /> <jats:tex-math>ptextup{-}q</jats:tex-math> </jats:alternatives> </jats:inline-formula> is an edge in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"normal\">Γ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0018_ineq_0001.png\" /> <jats:tex-math>Gamma(G)</jats:tex-math> </jats:alternatives> </jats:inline-formula> if and only if 𝐺 has an element of order <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:mi>q</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0018_ineq_0005.png\" /> <jats:tex-math>pq</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Since their inception in the 1970s, these graphs have been studied extensively; however, completely classifying the possible prime graphs for larger families of groups remains a difficult problem. For solvable groups, such a classification was found in 2015. In this paper, we go beyond solvable groups for the first time and characterize the prime graphs of a more general class of groups we call pseudo-solvable. These are groups whose composition factors are either cyclic or isomorphic to <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>A</m:mi> <m:mn>5</m:mn> </m:msub> </m:math> <jats:inline-graphic xmln","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138504127","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}