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.
{"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}
Francesco de Giovanni, Marco Trombetti, Bertram A. F. Wehrfritz
Abstract A well-known theorem of Philip Hall states that if a group 𝐺 has a nilpotent normal subgroup 𝑁 such that G/N′ 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 G/N′ 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.
Philip Hall的一个著名定理指出，如果一个群𝐺有一个幂零的正子群倘使G/N′G/N^{素数}是幂零的，则𝐺本身也是幂零的。我们说一个群类𝔛是一个霍尔类，如果它包含所有群𝐺承认一个幂零的正则子群(即G/N ' G/N^{素数}属于𝔛)。[F]中已经给出了例子。de Giovanni, M. Trombetti和B. a . F. Wehfritz，群的霍尔类，以显示:对于霍尔类的许多自然选择𝔛，有限的-𝔛群不能形成霍尔类。虽然这些例子通常是线性的，但我们在这里的目的是证明在线性群的某些自然子类中情况要好得多。
{"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}
Abstract Let Fq mathbb{F}_{q} be the finite field with 𝑞 elements and consider the 𝑛-dimensional Fq mathbb{F}_{q} -vector space V=Fqn V=mathbb{F}_{q}^{n} . In this paper, we define a closure operator on the subgroup lattice of the group G=PGL(V) 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 μ(H,G)≠0 mu(H,G)neq 0 . Moreover, we establish a polynomial bound on the number c(m) 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 GL(V) mathrm{GL}(V) and the same results proven for this group.
摘要设F q mathbb{F} ＿q{为具有𝑞元的有限域，考虑𝑛-dimensional F q }mathbb{F} ＿q{ -向量空间V= F q n V= }mathbb{F} ＿q{^}n{。本文在群G= PGL²(V) G= }mathrm{PGL} (V)的子群格上定义了一个闭包算子。令其表示这个格的Möbius函数。目的是使用这个闭包算子来描述𝐺的子群𝐻，其中μ≠(H,G)≠0 mu (H,G) neq 0。此外，我们在𝐺中建立了指标𝑚的闭子群𝐻的数c¹(m) c(m)的多项式界，其中𝐻-invariant子空间的格同构于链的乘积。这个边界只取决于𝑚，而不取决于𝑛和𝑞的选择。通过考虑GL ＿ (V) mathrm{GL} (V)的子群格的类似闭包算子，得到了同样的结果。
{"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}
Francesca Dalla Volta, Fabio Mastrogiacomo, Pablo Spiga
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.
{"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}
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.
{"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}
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 CAT(0) mathrm{CAT}(0) , C(6) 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 C(4) C(4) - T(4) T(4) graphical complexes of groups and outline their basic properties.
{"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}
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.
摘要利用二阶有限的𝑝-group𝐺对奇数素数𝑝的纯态，考虑了多重纯态的商群T¹(G) T(G)。通过第一作者的工作，我们知道T(G) T(G)包含一个p r−1 (p−1)p^{r-1}(p-1)阶的循环子群，其中p r p^{r}是𝐺的商的中心的指数。在本文中，我们将展示𝐺(r=1 r=1)的例子，使得T¹(G) T(G)的阶恰好是p−1 p-1，这是尽可能小的。
{"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}
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.
{"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}
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.
{"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}
Ziyu Huang, Thomas Michael Keller, Shane Kissinger, Wen Plotnick, Maya Roma, Yong Yang
The prime graph Γ(G)Gamma(G) of a finite group 𝐺 (also known as the Gruenberg–Kegel graph) has as its vertices the prime divisors of |G|lvert Grvert, and p-qptextup{-}q is an edge in Γ(G)Gamma(G) if and only if 𝐺 has an element of order pqpq. 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 A5
有限群𝐺(也称为Gruenberg-Kegel图)的质数图Γ (G) Gamma (G)的顶点是| G | lvert G rvert的质数因子，p¹-q ptextup{-q}是Γ (G) Gamma (G)中的一条边，当且仅当𝐺有一个p¹q pq阶的元素。自20世纪70年代出现以来，这些图表得到了广泛的研究;然而，对于较大群族的可能素图的完全分类仍然是一个难题。对于可解群，这种分类是在2015年发现的。在本文中，我们第一次超越了可解群，并刻画了一类更一般的群的素图，我们称之为伪可解群。这些群的组成因子是循环的或与a5 {A＿5}同构的。分类基于两个条件:顶点{2,3,5｛2,3,5}｝在Γ (G) overline{Gamma} (G)中形成三角形，或{p,3,5 ｛p,3,5}｝在某些素数p≠2 p neq 2中形成三角形。本文发展的思想也为今后对更一般的有限群的素数图进行分类和分析奠定了基础。
{"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}
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