Abstract One version of Whitehead’s famous cut vertex lemma says that if an element of a free group is part of a free basis, then a certain graph associated to its conjugacy class that we call the star graph is either disconnected or has a cut vertex. We state and prove a version of this lemma for conjugacy classes of elements and convex-cocompact subgroups of groups acting cocompactly on trees with finitely generated edge stabilizers.
{"title":"On Whitehead’s cut vertex lemma","authors":"Rylee Alanza Lyman","doi":"10.1515/jgth-2022-0089","DOIUrl":"https://doi.org/10.1515/jgth-2022-0089","url":null,"abstract":"Abstract One version of Whitehead’s famous cut vertex lemma says that if an element of a free group is part of a free basis, then a certain graph associated to its conjugacy class that we call the star graph is either disconnected or has a cut vertex. We state and prove a version of this lemma for conjugacy classes of elements and convex-cocompact subgroups of groups acting cocompactly on trees with finitely generated edge stabilizers.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"56 1","pages":"665 - 675"},"PeriodicalIF":0.5,"publicationDate":"2022-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81128153","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 𝜎 be a partition of the set of all primes, and let 𝔉 denote a hereditary formation. We describe all formations 𝔉 for which the 𝔉-hypercenter and the intersection of weak 𝐾-𝔉-subnormalizers of all Sylow subgroups coincide in every finite group. In particular, the formation of all 𝜎-nilpotent groups has this property. With the help of our results, we solve a particular case of Shemetkov’s problem about the intersection of 𝔉-maximal subgroups and the 𝔉-hypercenter. As a corollary, we obtain Hall’s classical result about the hypercenter. We prove that the non-𝜎-nilpotent graph of a group is connected and its diameter is at most 3.
{"title":"On the 𝜎-nilpotent hypercenter of finite groups","authors":"V. I. Murashka, A. Vasil'ev","doi":"10.1515/jgth-2021-0138","DOIUrl":"https://doi.org/10.1515/jgth-2021-0138","url":null,"abstract":"Abstract Let 𝜎 be a partition of the set of all primes, and let 𝔉 denote a hereditary formation. We describe all formations 𝔉 for which the 𝔉-hypercenter and the intersection of weak 𝐾-𝔉-subnormalizers of all Sylow subgroups coincide in every finite group. In particular, the formation of all 𝜎-nilpotent groups has this property. With the help of our results, we solve a particular case of Shemetkov’s problem about the intersection of 𝔉-maximal subgroups and the 𝔉-hypercenter. As a corollary, we obtain Hall’s classical result about the hypercenter. We prove that the non-𝜎-nilpotent graph of a group is connected and its diameter is at most 3.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"83 1","pages":"1083 - 1098"},"PeriodicalIF":0.5,"publicationDate":"2022-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79975569","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 study a certain family of simple fusion systems over finite 3-groups, ones that involve Todd modules of the Mathieu groups 2 M 12 2M_{12} , M 11 M_{11} , and A 6 = O 2 ( M 10 ) A_{6}=O^{2}(M_{10}) over F 3 mathbb{F}_{3} , and show that they are all isomorphic to the 3-fusion systems of almost simple groups. As one consequence, we give new 3-local characterizations of Conway’s sporadic simple groups.
{"title":"Fusion systems realizing certain Todd modules","authors":"B. Oliver","doi":"10.1515/jgth-2022-0074","DOIUrl":"https://doi.org/10.1515/jgth-2022-0074","url":null,"abstract":"Abstract We study a certain family of simple fusion systems over finite 3-groups, ones that involve Todd modules of the Mathieu groups 2 M 12 2M_{12} , M 11 M_{11} , and A 6 = O 2 ( M 10 ) A_{6}=O^{2}(M_{10}) over F 3 mathbb{F}_{3} , and show that they are all isomorphic to the 3-fusion systems of almost simple groups. As one consequence, we give new 3-local characterizations of Conway’s sporadic simple groups.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"25 1","pages":"421 - 505"},"PeriodicalIF":0.5,"publicationDate":"2022-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84069989","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 show that if a nontrivial group admits a locally invariant ordering, then it admits uncountably many locally invariant orderings. For the case of a left-orderable group, we provide an explicit construction of uncountable families of locally invariant orderings; for a general group, we provide an existence theorem that applies compactness to yield uncountably many locally invariant orderings. Along the way, we define and investigate the space of locally invariant orderings of a group, the natural group actions on this space, and their relationship to the space of left-orderings.
{"title":"The number of locally invariant orderings of a group","authors":"I. Ba, A. Clay, I. Thompson","doi":"10.1515/jgth-2022-0126","DOIUrl":"https://doi.org/10.1515/jgth-2022-0126","url":null,"abstract":"Abstract We show that if a nontrivial group admits a locally invariant ordering, then it admits uncountably many locally invariant orderings. For the case of a left-orderable group, we provide an explicit construction of uncountable families of locally invariant orderings; for a general group, we provide an existence theorem that applies compactness to yield uncountably many locally invariant orderings. Along the way, we define and investigate the space of locally invariant orderings of a group, the natural group actions on this space, and their relationship to the space of left-orderings.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"43 1","pages":"1003 - 1021"},"PeriodicalIF":0.5,"publicationDate":"2022-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80812269","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 p k p^{k} a prime power dividing | G | lvert Grvert . A subgroup 𝐻 of 𝐺 is said to be ℳ-supplemented in 𝐺 if there exists a subgroup 𝐾 of 𝐺 such that G = H K G=HK and H i K < G H_{i}K
摘要设𝐺为有限群，p k p^{k}为素数幂除以| G | lvert G rvert。如果𝐺存在一个子群𝐾，使得对于𝐻的每一个极大子群H i H＿i, G=H∑K G=HK且H i∑K
{"title":"On ℳ-supplemented subgroups","authors":"Yuedi Zeng","doi":"10.1515/jgth-2021-0195","DOIUrl":"https://doi.org/10.1515/jgth-2021-0195","url":null,"abstract":"Abstract Let 𝐺 be a finite group and p k p^{k} a prime power dividing | G | lvert Grvert . A subgroup 𝐻 of 𝐺 is said to be ℳ-supplemented in 𝐺 if there exists a subgroup 𝐾 of 𝐺 such that G = H K G=HK and H i K < G H_{i}K<G for every maximal subgroup H i H_{i} of 𝐻. In this paper, we complete the classification of the finite groups 𝐺 in which all subgroups of order p k p^{k} are ℳ-supplemented. In particular, we show that if k ≥ 2 kgeq 2 , then G / O p ′ ( G ) G/mathbf{O}_{p^{prime}}(G) is supersolvable with a normal Sylow 𝑝-subgroup and a cyclic 𝑝-complement.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"11 1","pages":"1099 - 1107"},"PeriodicalIF":0.5,"publicationDate":"2022-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84174199","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 compute powers in the wreath product G ≀ S n Gwr S_{n} for any finite group 𝐺. For r ≥ 2 rgeq 2 a prime, consider ω r : G ≀ S n → G ≀ S n omega_{r}colon Gwr S_{n}to Gwr S_{n} defined by g ↦ g r gmapsto g^{r} . Let P r ( G ≀ S n ) := | ω r ( G ≀ S n ) | | G | n n ! P_{r}(Gwr S_{n}):=frac{lvertomega_{r}(Gwr S_{n})rvert}{lvert Grvert^{n}n!} be the probability that a randomly chosen element in G ≀ S n Gwr S_{n} is an 𝑟-th power. We prove P r ( G ≀ S n + 1 ) = P r ( G ≀ S n ) P_{r}(Gwr S_{n+1})=P_{r}(Gwr S_{n}) for all n ≢ - 1 ( mod r ) nnotequiv-1 (mathrm{mod} r) if the order of 𝐺 is coprime to 𝑟. We also give a formula for the number of conjugacy classes that are 𝑟-th powers in G ≀ S n Gwr S_{n} .
抽象的这篇文章,我们《wreath鲍尔compute广告G≀S n G wr S_{}对于任何有限的𝐺集团。为r≥2 r geq a prime,认为ωr: G≀S n→G≀结肠G n omega_ {r的wr S_ {n}到G wr S_ (n):是由G↦G r G r mapsto G ^{}。让P r S(G≀n): = |ωS r(G≀n) | | G | nn !P_ {r} (G n wr S_ {}): = frac {lvert r omega_ {} (G wr S_ {n}) rvert} {lvert G rvert ^ {n, n !be a probability那randomly被选中元素》是G≀S n G wr S_{}是一个𝑟-th电源。我们证明P r S(G≀n + 1) = P r S(G≀n) P_ {r} (G wr S_ (n + 1)) = r P_ {} (G wr S_ {n})为所有n≢- 1(modr) n equiv-1音符(mathrm {mod} r)如果《𝐺是coprime到𝑟勋章。我们当家》也给a配方for conjugacy课堂这是鲍尔𝑟-th in G≀S n G wr S_{}。
{"title":"Powers in wreath products of finite groups","authors":"Rijubrata Kundu, Sudipa Mondal","doi":"10.1515/jgth-2021-0057","DOIUrl":"https://doi.org/10.1515/jgth-2021-0057","url":null,"abstract":"Abstract In this paper, we compute powers in the wreath product G ≀ S n Gwr S_{n} for any finite group 𝐺. For r ≥ 2 rgeq 2 a prime, consider ω r : G ≀ S n → G ≀ S n omega_{r}colon Gwr S_{n}to Gwr S_{n} defined by g ↦ g r gmapsto g^{r} . Let P r ( G ≀ S n ) := | ω r ( G ≀ S n ) | | G | n n ! P_{r}(Gwr S_{n}):=frac{lvertomega_{r}(Gwr S_{n})rvert}{lvert Grvert^{n}n!} be the probability that a randomly chosen element in G ≀ S n Gwr S_{n} is an 𝑟-th power. We prove P r ( G ≀ S n + 1 ) = P r ( G ≀ S n ) P_{r}(Gwr S_{n+1})=P_{r}(Gwr S_{n}) for all n ≢ - 1 ( mod r ) nnotequiv-1 (mathrm{mod} r) if the order of 𝐺 is coprime to 𝑟. We also give a formula for the number of conjugacy classes that are 𝑟-th powers in G ≀ S n Gwr S_{n} .","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"30 1","pages":"941 - 964"},"PeriodicalIF":0.5,"publicationDate":"2022-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76131877","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 𝐾 and 𝐷 be conjugacy classes of a finite group 𝐺, and suppose that we have K n = D ∪ D - 1 K^{n}=Dcup D^{-1} for some integer n ≥ 2 ngeq 2 . Under these assumptions, it was conjectured that ⟨ K ⟩ langle Krangle must be a (normal) solvable subgroup of 𝐺. Recently R. D. Camina has demonstrated that the conjecture is valid for any n ≥ 4 ngeq 4 , and this is done by applying combinatorial results, the main of which concerns subsets with small doubling in a finite group. In this note, we solve the case n = 3 n=3 by appealing to other combinatorial results, such as an estimate of the cardinality of the product of two normal sets in a finite group as well as to some recent techniques and theorems.
摘要设𝐾和𝐷是有限群𝐺的共轭类，并设K n=D∪D -1 K^{n}=D cup D^{-1}对于某整数n≥2 n geq 2。在这些假设下，我们推测⟨K⟩langle K rangle必须是𝐺的一个(正规的)可解的子群。最近研发。Camina已经证明了这个猜想对任何n≥4 n geq 4都是有效的，这是通过应用组合结果来完成的，其中主要涉及有限群中具有小倍的子集。在这篇笔记中，我们通过求助于其他的组合结果来解决n= 3n =3的情况，例如有限群中两个正态集积的基数的估计，以及一些最新的技术和定理。
{"title":"On powers of conjugacy classes in finite groups","authors":"A. Beltrán","doi":"10.1515/jgth-2021-0156","DOIUrl":"https://doi.org/10.1515/jgth-2021-0156","url":null,"abstract":"Abstract Let 𝐾 and 𝐷 be conjugacy classes of a finite group 𝐺, and suppose that we have K n = D ∪ D - 1 K^{n}=Dcup D^{-1} for some integer n ≥ 2 ngeq 2 . Under these assumptions, it was conjectured that ⟨ K ⟩ langle Krangle must be a (normal) solvable subgroup of 𝐺. Recently R. D. Camina has demonstrated that the conjecture is valid for any n ≥ 4 ngeq 4 , and this is done by applying combinatorial results, the main of which concerns subsets with small doubling in a finite group. In this note, we solve the case n = 3 n=3 by appealing to other combinatorial results, such as an estimate of the cardinality of the product of two normal sets in a finite group as well as to some recent techniques and theorems.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"46 1","pages":"965 - 971"},"PeriodicalIF":0.5,"publicationDate":"2022-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90795215","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 investigate the Tits alternative for cyclically presented groups with length-four positive relators in terms of a system of congruences (A), (B), (C) in the defining parameters, introduced by Bogley and Parker. Except for the case when (B) holds and neither (A) nor (C) hold, we show that the Tits alternative is satisfied; in the remaining case, we show that the Tits alternative is satisfied when the number of generators of the cyclic presentation is at most 20.
{"title":"On the Tits alternative for cyclically presented groups with length-four positive relators","authors":"Shaun Isherwood, Gerald Williams","doi":"10.1515/jgth-2021-0131","DOIUrl":"https://doi.org/10.1515/jgth-2021-0131","url":null,"abstract":"Abstract We investigate the Tits alternative for cyclically presented groups with length-four positive relators in terms of a system of congruences (A), (B), (C) in the defining parameters, introduced by Bogley and Parker. Except for the case when (B) holds and neither (A) nor (C) hold, we show that the Tits alternative is satisfied; in the remaining case, we show that the Tits alternative is satisfied when the number of generators of the cyclic presentation is at most 20.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"20 1 1","pages":"837 - 850"},"PeriodicalIF":0.5,"publicationDate":"2022-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74186771","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 and 𝐻 be residually nilpotent groups. Then 𝐺 and 𝐻 are in the same nilpotent genus if they have the same lower central quotients (up to isomorphism). A potentially stronger condition is that 𝐻 is para-𝐺 if there exists a monomorphism of 𝐺 into 𝐻 which induces isomorphisms between the corresponding quotients of their lower central series. We first consider finitely generated residually nilpotent groups and find sufficient conditions on the monomorphism so that 𝐻 is para-𝐺. We then prove that, for certain polycyclic groups, if 𝐻 is para-𝐺, then 𝐺 and 𝐻 have the same Hirsch length. We also prove that the pro-nilpotent completions of these polycyclic groups are locally polycyclic.
{"title":"The nilpotent genus of finitely generated residually nilpotent groups","authors":"N. O’Sullivan","doi":"10.1515/jgth-2022-0098","DOIUrl":"https://doi.org/10.1515/jgth-2022-0098","url":null,"abstract":"Abstract Let 𝐺 and 𝐻 be residually nilpotent groups. Then 𝐺 and 𝐻 are in the same nilpotent genus if they have the same lower central quotients (up to isomorphism). A potentially stronger condition is that 𝐻 is para-𝐺 if there exists a monomorphism of 𝐺 into 𝐻 which induces isomorphisms between the corresponding quotients of their lower central series. We first consider finitely generated residually nilpotent groups and find sufficient conditions on the monomorphism so that 𝐻 is para-𝐺. We then prove that, for certain polycyclic groups, if 𝐻 is para-𝐺, then 𝐺 and 𝐻 have the same Hirsch length. We also prove that the pro-nilpotent completions of these polycyclic groups are locally polycyclic.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"25 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91257143","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 recent work, Cameron, Manna and Mehatari have studied the finite groups whose power graph is a cograph, which we refer to as power-cograph groups. They classify the nilpotent groups with this property, and they establish partial results in the general setting, highlighting certain number-theoretic difficulties that arise for the simple groups of the form PSL 2 ( q ) operatorname{PSL}_{2}(q) or Sz ( 2 2 e + 1 ) operatorname{Sz}(2^{2e+1}) . In this paper, we prove that these number-theoretic problems are in fact the only obstacles to the classification of non-solvable power-cograph groups. Specifically, for the non-solvable case, we give a classification of power-cograph groups in terms of such groups isomorphic to PSL 2 ( q ) operatorname{PSL}_{2}(q) or Sz ( 2 2 e + 1 ) operatorname{Sz}(2^{2e+1}) . For the solvable case, we are able to precisely describe the structure of solvable power-cograph groups. We obtain a complete classification of solvable power-cograph groups whose Gruenberg–Kegel graph is connected. Moreover, we reduce the case where the Gruenberg–Kegel graph is disconnected to the classification of 𝑝-groups admitting fixed-point-free automorphisms of prime power order, which is in general an open problem.
{"title":"Classification of non-solvable groups whose power graph is a cograph","authors":"Jendrik Brachter, Eda Kaja","doi":"10.1515/jgth-2022-0081","DOIUrl":"https://doi.org/10.1515/jgth-2022-0081","url":null,"abstract":"Abstract In recent work, Cameron, Manna and Mehatari have studied the finite groups whose power graph is a cograph, which we refer to as power-cograph groups. They classify the nilpotent groups with this property, and they establish partial results in the general setting, highlighting certain number-theoretic difficulties that arise for the simple groups of the form PSL 2 ( q ) operatorname{PSL}_{2}(q) or Sz ( 2 2 e + 1 ) operatorname{Sz}(2^{2e+1}) . In this paper, we prove that these number-theoretic problems are in fact the only obstacles to the classification of non-solvable power-cograph groups. Specifically, for the non-solvable case, we give a classification of power-cograph groups in terms of such groups isomorphic to PSL 2 ( q ) operatorname{PSL}_{2}(q) or Sz ( 2 2 e + 1 ) operatorname{Sz}(2^{2e+1}) . For the solvable case, we are able to precisely describe the structure of solvable power-cograph groups. We obtain a complete classification of solvable power-cograph groups whose Gruenberg–Kegel graph is connected. Moreover, we reduce the case where the Gruenberg–Kegel graph is disconnected to the classification of 𝑝-groups admitting fixed-point-free automorphisms of prime power order, which is in general an open problem.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"714 1","pages":"851 - 872"},"PeriodicalIF":0.5,"publicationDate":"2022-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76944166","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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