The aim of our paper is to generalize the tangent prolongation of Lie groups to non-associative multiplications and to examine how the weak associative and weak inverse properties are transferred to the multiplication defined on the tangent bundle. We obtain that the tangent prolongation of a $mathcal{C}^r$-differentiable loop ($rgeq 1$) is a $mathcal{C}^{r-1}$-differentiable loop that acquires the classical weak inverse and weak associative properties of the initial loop.
{"title":"Tangent prolongation of $mathcal{C}^r$-differentiable loops","authors":"'Agota Figula, P. Nagy","doi":"10.5486/pmd.2020.8818","DOIUrl":"https://doi.org/10.5486/pmd.2020.8818","url":null,"abstract":"The aim of our paper is to generalize the tangent prolongation of Lie groups to non-associative multiplications and to examine how the weak associative and weak inverse properties are transferred to the multiplication defined on the tangent bundle. We obtain that the tangent prolongation of a $mathcal{C}^r$-differentiable loop ($rgeq 1$) is a $mathcal{C}^{r-1}$-differentiable loop that acquires the classical weak inverse and weak associative properties of the initial loop.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"56 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72679486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-02-02DOI: 10.30504/JIMS.2020.107524
M. Vaughan-Lee
G.E. Wall gave two different proofs of a remarkable result about the multilinear Lie relators satisfied by groups of prime power exponent $q$. He showed that if $q$ is a power of the prime $p$, and if $f$ is a multilinear Lie relator in $n$ variables where $nneq1operatorname{mod}(p-1)$, then $f=0$ is a consequence of multilinear Lie relators in fewer than $n$ variables. For years I have struggled to understand his proofs, and while I still have not the slightest clue about his first proof published in the Journal of Algebra, I finally have some understanding of his second proof published in a conference proceedings. In this note I offer my insights into Wall's second proof of this theorem.
{"title":"Understanding Wall's theorem on dependence of Lie relators in Burnside groups","authors":"M. Vaughan-Lee","doi":"10.30504/JIMS.2020.107524","DOIUrl":"https://doi.org/10.30504/JIMS.2020.107524","url":null,"abstract":"G.E. Wall gave two different proofs of a remarkable result about the multilinear Lie relators satisfied by groups of prime power exponent $q$. He showed that if $q$ is a power of the prime $p$, and if $f$ is a multilinear Lie relator in $n$ variables where $nneq1operatorname{mod}(p-1)$, then $f=0$ is a consequence of multilinear Lie relators in fewer than $n$ variables. For years I have struggled to understand his proofs, and while I still have not the slightest clue about his first proof published in the Journal of Algebra, I finally have some understanding of his second proof published in a conference proceedings. In this note I offer my insights into Wall's second proof of this theorem.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82903675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-30DOI: 10.22108/IJGT.2020.119123.1574
Delaram Kahrobaei, M. Noce
The theory of Engel groups plays an important role in group theory since these groups are closely related to the Burnside problems. In this survey we consider several classical and novel algorithmic problems for Engel groups and propose several open problems. We study these problems with a view towards applications to cryptography.
{"title":"Algorithmic problems in Engel groups and cryptographic applications.","authors":"Delaram Kahrobaei, M. Noce","doi":"10.22108/IJGT.2020.119123.1574","DOIUrl":"https://doi.org/10.22108/IJGT.2020.119123.1574","url":null,"abstract":"The theory of Engel groups plays an important role in group theory since these groups are closely related to the Burnside problems. In this survey we consider several classical and novel algorithmic problems for Engel groups and propose several open problems. We study these problems with a view towards applications to cryptography.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"56 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77251602","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This is an expository book on unitary representations of topological groups, and of several dual spaces, which are spaces of such representations up to some equivalence. The most important notions are defined for topological groups, but a special attention is paid to the case of discrete groups. The unitary dual of a group $G$ is the space of equivalence classes of its irreducible unitary representations; it is both a topological space and a Borel space. The primitive dual is the space of weak equivalence classes of unitary irreducible representations. The normal quasi-dual is the space of quasi-equivalence classes of traceable factor representations; it is parametrized by characters, which can be finite or infinite. The theory is systematically illustrated by a series of specific examples: Heisenberg groups, affine groups of infinite fields, solvable Baumslag-Solitar groups, lamplighter groups, and general linear groups. Operator algebras play an important role in the exposition, in particular the von Neumann algebras associated to a unitary representation and C*-algebras associated to a locally compact group.
{"title":"Unitary Representations of Groups, Duals, and\u0000 Characters","authors":"Bachir Bekka, P. Harpe","doi":"10.1090/SURV/250","DOIUrl":"https://doi.org/10.1090/SURV/250","url":null,"abstract":"This is an expository book on unitary representations of topological groups, and of several dual spaces, which are spaces of such representations up to some equivalence. The most important notions are defined for topological groups, but a special attention is paid to the case of discrete groups. \u0000The unitary dual of a group $G$ is the space of equivalence classes of its irreducible unitary representations; it is both a topological space and a Borel space. The primitive dual is the space of weak equivalence classes of unitary irreducible representations. The normal quasi-dual is the space of quasi-equivalence classes of traceable factor representations; it is parametrized by characters, which can be finite or infinite. \u0000The theory is systematically illustrated by a series of specific examples: Heisenberg groups, affine groups of infinite fields, solvable Baumslag-Solitar groups, lamplighter groups, and general linear groups. \u0000Operator algebras play an important role in the exposition, in particular the von Neumann algebras associated to a unitary representation and C*-algebras associated to a locally compact group.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"100 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74177491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We determine the ranks of string C-group representations of the groups ${rm PSp}(4,mathbb{F}_q)congOmega(5,mathbb{F}_q)$, and comment on those of higher-dimensional symplectic and orthogonal groups.
{"title":"On the ranks of string C-group\u0000 representations for symplectic and orthogonal\u0000 groups","authors":"P. Brooksbank","doi":"10.1090/CONM/764/15329","DOIUrl":"https://doi.org/10.1090/CONM/764/15329","url":null,"abstract":"We determine the ranks of string C-group representations of the groups ${rm PSp}(4,mathbb{F}_q)congOmega(5,mathbb{F}_q)$, and comment on those of higher-dimensional symplectic and orthogonal groups.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74266052","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-11-26DOI: 10.22108/IJGT.2020.124183.1641
E. Campedel, A. Caranti, I. D. Corso
We record for reference a detailed description of the automorphism groups of the groups of order $p^{2} q$, where $p$ and $q$ are distinct primes.
我们记录了$p^{2} q$阶群的自同构群的详细描述，其中$p$和$q$是不同的素数。
{"title":"The automorphism groups of groups of order $p^{2} q$","authors":"E. Campedel, A. Caranti, I. D. Corso","doi":"10.22108/IJGT.2020.124183.1641","DOIUrl":"https://doi.org/10.22108/IJGT.2020.124183.1641","url":null,"abstract":"We record for reference a detailed description of the automorphism groups of the groups of order $p^{2} q$, where $p$ and $q$ are distinct primes.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82573838","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct new examples of groups with cohomological dimension 2 and geometric dimension 3 with respect to the families of finite subgroups, virtually abelian groups of bounded rank, and the family of virtually poly-cyclic subgroups. Our main ingredients are the examples constructed by Brady-Leary-Nucinckis and Fluch-Leary, and Bass-Serre theory.
{"title":"Groups acting on trees and the Eilenberg–Ganea problem for families","authors":"Luis Jorge S'anchez Saldana","doi":"10.1090/proc/15203","DOIUrl":"https://doi.org/10.1090/proc/15203","url":null,"abstract":"We construct new examples of groups with cohomological dimension 2 and geometric dimension 3 with respect to the families of finite subgroups, virtually abelian groups of bounded rank, and the family of virtually poly-cyclic subgroups. Our main ingredients are the examples constructed by Brady-Leary-Nucinckis and Fluch-Leary, and Bass-Serre theory.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81694363","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the present paper we prove a weak form of sandwich classification for the overgroups of the subsystem subgroup $E(Delta,R)$ of the Chevalley group $G(Phi,R)$ where $Phi$ is a symply laced root sysetem and $Delta$ is its sufficiently large subsystem. Namely we show that for any such an overgroup $H$ there exists a unique net of ideals $sigma$ of the ring $R$ such that $E(Phi,Delta,R,sigma)le Hle {mathop{mathrm{Stab}}nolimits}_{G(Phi,R)}(L(sigma))$ where $E(Phi,Delta,R,sigma)$ is an elementary subgroup associated with the net and $L(sigma)$ is a corresponding subalgebra of the Chevalley Lie algebra.
{"title":"Overgroups of subsystem subgroups in exceptional groups: 2A1-proof","authors":"P. Gvozdevsky","doi":"10.1090/spmj/1682","DOIUrl":"https://doi.org/10.1090/spmj/1682","url":null,"abstract":"In the present paper we prove a weak form of sandwich classification for the overgroups of the subsystem subgroup $E(Delta,R)$ of the Chevalley group $G(Phi,R)$ where $Phi$ is a symply laced root sysetem and $Delta$ is its sufficiently large subsystem. Namely we show that for any such an overgroup $H$ there exists a unique net of ideals $sigma$ of the ring $R$ such that $E(Phi,Delta,R,sigma)le Hle {mathop{mathrm{Stab}}nolimits}_{G(Phi,R)}(L(sigma))$ where $E(Phi,Delta,R,sigma)$ is an elementary subgroup associated with the net and $L(sigma)$ is a corresponding subalgebra of the Chevalley Lie algebra.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"94 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91066692","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $hat{F}$ be a free pro-$p$ non-abelian group, and let $Delta$ be a local commutative complete ring with a maximal ideal $I$ such that $textrm{char}(Delta/I)=p$. In [Zu], Zubkov showed that when $pneq2$, the pro-$p$ congruence subgroup $GL_{2}^{1}(Delta)=ker(GL_{2}(Delta)overset{DeltatoDelta/I}{longrightarrow}GL_{2}(Delta/I))$ admits a pro-$p$ identity. I.e. there exists an element $1neq winhat{F}$ that vanishes under any continuous homomorphism $hat{F}to GL_{2}^{1}(Delta)$. In this paper we investigate the case $p=2$. The main result is that when $textrm{char}(Delta)=2$, the pro-$2$ group $GL_{2}^{1}(Delta)$ admits a pro-$2$ identity. This result was obtained by the use of trace identities that are originated in PI-theory.
{"title":"On pro-$2$ identities of $2times 2$ linear groups","authors":"David el-Chai Ben-Ezra, E. Zelmanov","doi":"10.1090/TRAN/8327","DOIUrl":"https://doi.org/10.1090/TRAN/8327","url":null,"abstract":"Let $hat{F}$ be a free pro-$p$ non-abelian group, and let $Delta$ be a local commutative complete ring with a maximal ideal $I$ such that $textrm{char}(Delta/I)=p$. In [Zu], Zubkov showed that when $pneq2$, the pro-$p$ congruence subgroup $GL_{2}^{1}(Delta)=ker(GL_{2}(Delta)overset{DeltatoDelta/I}{longrightarrow}GL_{2}(Delta/I))$ admits a pro-$p$ identity. I.e. there exists an element $1neq winhat{F}$ that vanishes under any continuous homomorphism $hat{F}to GL_{2}^{1}(Delta)$. \u0000In this paper we investigate the case $p=2$. The main result is that when $textrm{char}(Delta)=2$, the pro-$2$ group $GL_{2}^{1}(Delta)$ admits a pro-$2$ identity. This result was obtained by the use of trace identities that are originated in PI-theory.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"110 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91324133","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-10-10DOI: 10.1016/J.JALGEBRA.2021.07.008
P. Shumyatsky, P. Zalesskii
{"title":"Profinite groups in which centralizers are virtually procyclic","authors":"P. Shumyatsky, P. Zalesskii","doi":"10.1016/J.JALGEBRA.2021.07.008","DOIUrl":"https://doi.org/10.1016/J.JALGEBRA.2021.07.008","url":null,"abstract":"","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81630052","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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