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The groups 𝐺 satisfying a functional equation 𝑓(𝑥𝑘) = 𝑥𝑓(𝑥) for some 𝑘 ∈ 𝐺 对于某些𝑘∈𝐺，群𝐺满足一个泛函方程𝑓(χ𝑘)= χ𝑓(χ)
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2022-05-12 DOI: 10.1515/jgth-2021-0158
Dominik Bernhardt, Tim Boykett, Alice Devillers, Johannes Flake, S. Glasby
{"title":"The groups 𝐺 satisfying a functional equation 𝑓(𝑥𝑘) = 𝑥𝑓(𝑥) for some 𝑘 ∈ 𝐺","authors":"Dominik Bernhardt, Tim Boykett, Alice Devillers, Johannes Flake, S. Glasby","doi":"10.1515/jgth-2021-0158","DOIUrl":"https://doi.org/10.1515/jgth-2021-0158","url":null,"abstract":"Abstract We study the groups 𝐺 with the curious property that there exists an element k ∈ G kin G and a function f : G → G fcolon Gto G such that f ⁢ ( x ⁢ k ) = x ⁢ f ⁢ ( x ) f(xk)=xf(x) holds for all x ∈ G xin G . This property arose from the study of near-rings and input-output automata on groups. We call a group with this property a 𝐽-group. Finite 𝐽-groups must have odd order, and hence are solvable. We prove that every finite nilpotent group of odd order is a 𝐽-group if its nilpotency class 𝑐 satisfies c ⩽ 6 cleqslant 6 . If 𝐺 is a finite 𝑝-group, with p > 2 p>2 and p 2 > 2 ⁢ c - 1 p^{2}>2c-1 , then we prove that 𝐺 is 𝐽-group. Finally, if p > 2 p>2 and 𝐺 is a regular 𝑝-group or, more generally, a power-closed one (i.e., in each section and for each m ⩾ 1 mgeqslant 1 , the subset of p m p^{m} -th powers is a subgroup), then we prove that 𝐺 is a 𝐽-group.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74740310","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}
引用次数: 0
On Whitehead’s cut vertex lemma 关于Whitehead的切顶点引理
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2022-05-12 DOI: 10.1515/jgth-2022-0089
Rylee Alanza Lyman
{"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":null,"pages":null},"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}
引用次数: 0
On the 𝜎-nilpotent hypercenter of finite groups 关于有限群的𝜎-nilpotent超中心
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2022-05-07 DOI: 10.1515/jgth-2021-0138
V. I. Murashka, A. Vasil'ev
{"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":null,"pages":null},"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}
引用次数: 2
Fusion systems realizing certain Todd modules 实现某些Todd模块的融合系统
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2022-04-16 DOI: 10.1515/jgth-2022-0074
B. Oliver
{"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":null,"pages":null},"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}
引用次数: 2
The number of locally invariant orderings of a group 群的局部不变序的个数
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2022-04-02 DOI: 10.1515/jgth-2022-0126
I. Ba, A. Clay, I. Thompson
{"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":null,"pages":null},"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}
引用次数: 0
On ℳ-supplemented subgroups 关于补充了tag的子组
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2022-03-26 DOI: 10.1515/jgth-2021-0195
Yuedi Zeng
{"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":null,"pages":null},"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}
引用次数: 0
Powers in wreath products of finite groups 有限群环积中的幂
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2022-03-23 DOI: 10.1515/jgth-2021-0057
Rijubrata Kundu, Sudipa Mondal
{"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":null,"pages":null},"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}
引用次数: 1
On powers of conjugacy classes in finite groups 有限群中共轭类的幂
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2022-03-17 DOI: 10.1515/jgth-2021-0156
A. Beltrán
{"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":null,"pages":null},"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}
引用次数: 1
On the Tits alternative for cyclically presented groups with length-four positive relators 长度为4的正相关群的周期性呈现的Tits替代
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2022-03-08 DOI: 10.1515/jgth-2021-0131
Shaun Isherwood, Gerald Williams
{"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":null,"pages":null},"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}
引用次数: 1
The nilpotent genus of finitely generated residually nilpotent groups 有限生成残幂零群的幂零格
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2022-03-04 DOI: 10.1515/jgth-2022-0098
N. O’Sullivan
{"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":null,"pages":null},"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}
引用次数: 0
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