Pub Date : 2020-07-16DOI: 10.14712/1213-7243.2020.043
Mark Greer, Lee Raney
Given a uniquely 2-divisible group $G$, we study a commutative loop $(G,circ)$ which arises as a result of a construction in cite{baer}. We investigate some general properties and applications of $circ$ and determine a necessary and sufficient condition on $G$ in order for $(G, circ)$ to be Moufang. In cite{greer14}, it is conjectured that $G$ is metabelian if and only if $(G, circ)$ is an automorphic loop. We answer a portion of this conjecture in the affirmative: in particular, we show that if $G$ is a split metabelian group of odd order, then $(G, circ)$ is automorphic.
{"title":"Automorphic loops and metabelian groups","authors":"Mark Greer, Lee Raney","doi":"10.14712/1213-7243.2020.043","DOIUrl":"https://doi.org/10.14712/1213-7243.2020.043","url":null,"abstract":"Given a uniquely 2-divisible group $G$, we study a commutative loop $(G,circ)$ which arises as a result of a construction in cite{baer}. We investigate some general properties and applications of $circ$ and determine a necessary and sufficient condition on $G$ in order for $(G, circ)$ to be Moufang. In cite{greer14}, it is conjectured that $G$ is metabelian if and only if $(G, circ)$ is an automorphic loop. We answer a portion of this conjecture in the affirmative: in particular, we show that if $G$ is a split metabelian group of odd order, then $(G, circ)$ is automorphic.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"308 ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91466131","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 : 2020-06-25DOI: 10.1142/S179352532150031X
J. Taback, Alden Walker
For an element in $BS(1,n) = langle t,a | tat^{-1} = a^n rangle$ written in the normal form $t^{-u}a^vt^w$ with $u,w geq 0$ and $v in mathbb{Z}$, we exhibit a geodesic word representing the element and give a formula for its word length with respect to the generating set ${t,a}$. Using this word length formula, we prove that there are sets of elements of positive density of positive, negative and zero conjugation curvature, as defined by Bar Natan, Duchin and Kropholler.
{"title":"Conjugation curvature in solvable Baumslag–Solitar groups","authors":"J. Taback, Alden Walker","doi":"10.1142/S179352532150031X","DOIUrl":"https://doi.org/10.1142/S179352532150031X","url":null,"abstract":"For an element in $BS(1,n) = langle t,a | tat^{-1} = a^n rangle$ written in the normal form $t^{-u}a^vt^w$ with $u,w geq 0$ and $v in mathbb{Z}$, we exhibit a geodesic word representing the element and give a formula for its word length with respect to the generating set ${t,a}$. Using this word length formula, we prove that there are sets of elements of positive density of positive, negative and zero conjugation curvature, as defined by Bar Natan, Duchin and Kropholler.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89067192","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 : 2020-06-22DOI: 10.14712/1213-7243.2020.039
N. Didurik, V. Shcherbacov
We prolong Kunen research about existence of units (left, right, two-sided) in quasigroups with classical Bol-Moufang type identities. These identities were listed in Fenvesh article.
{"title":"Units in quasigroups with classical Bol--Moufang type identities","authors":"N. Didurik, V. Shcherbacov","doi":"10.14712/1213-7243.2020.039","DOIUrl":"https://doi.org/10.14712/1213-7243.2020.039","url":null,"abstract":"We prolong Kunen research about existence of units (left, right, two-sided) in quasigroups with classical Bol-Moufang type identities. These identities were listed in Fenvesh article.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"169 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74895628","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 : 2020-06-17DOI: 10.33048/SEMI.2020.17.087
D. Revin, A. Zavarnitsine
We explore the extent to which constructing the inductive theory of $mathfrak{X}$-submaximal subgroups is possible. To this end, we study the behavior of $pi$-submaximal subgroups under homomorphisms with $pi$-separable kernels and construct examples where such behavior is irregular.
{"title":"The behavior of $pi$-submaximal subgroups under homomorphisms with $pi$-separable kernels","authors":"D. Revin, A. Zavarnitsine","doi":"10.33048/SEMI.2020.17.087","DOIUrl":"https://doi.org/10.33048/SEMI.2020.17.087","url":null,"abstract":"We explore the extent to which constructing the inductive theory of $mathfrak{X}$-submaximal subgroups is possible. To this end, we study the behavior of $pi$-submaximal subgroups under homomorphisms with $pi$-separable kernels and construct examples where such behavior is irregular.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79358289","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 : 2020-06-15DOI: 10.1007/S00605-020-01484-7
K. Dekimpe, Daciberg Lima Gonçalves, Oscar Ocampo
{"title":"The $$R_infty $$ property for pure Artin braid groups","authors":"K. Dekimpe, Daciberg Lima Gonçalves, Oscar Ocampo","doi":"10.1007/S00605-020-01484-7","DOIUrl":"https://doi.org/10.1007/S00605-020-01484-7","url":null,"abstract":"","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"84 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74284540","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 : 2020-06-02DOI: 10.4007/ANNALS.2021.193.2.5
Timothy C. Burness, R. Guralnick, Scott Harper
A group $G$ is said to be $frac{3}{2}$-generated if every nontrivial element belongs to a generating pair. It is easy to see that if $G$ has this property then every proper quotient of $G$ is cyclic. In this paper we prove that the converse is true for finite groups, which settles a conjecture of Breuer, Guralnick and Kantor from 2008. In fact, we prove a much stronger result, which solves a problem posed by Brenner and Wiegold in 1975. Namely, if $G$ is a finite group and every proper quotient of $G$ is cyclic, then for any pair of nontrivial elements $x_1,x_2 in G$, there exists $y in G$ such that $G = langle x_1, y rangle = langle x_2, y rangle$. In other words, $s(G) geqslant 2$, where $s(G)$ is the spread of $G$. Moreover, if $u(G)$ denotes the more restrictive uniform spread of $G$, then we can completely characterise the finite groups $G$ with $u(G) = 0$ and $u(G)=1$. To prove these results, we first establish a reduction to almost simple groups. For simple groups, the result was proved by Guralnick and Kantor in 2000 using probabilistic methods and since then the almost simple groups have been the subject of several papers. By combining our reduction theorem and this earlier work, it remains to handle the groups whose socles are exceptional groups of Lie type and this is the case we treat in this paper.
如果每个非平凡元素都属于生成对，则称群$G$是$frac{3}{2}$生成的。很容易看出，如果$G$具有这个性质，那么$G$的每一个固有商都是循环的。本文证明了有限群的逆命题成立，从而解决了Breuer、Guralnick和Kantor在2008年提出的一个猜想。事实上，我们证明了一个更强的结果，它解决了Brenner和Wiegold在1975年提出的一个问题。即，如果$G$是一个有限群，且$G$的每一个真商都是循环的，则对于任意一对非平凡元素$x_1,x_2 in G$，存在$y in G$使得$G = langle x_1, y rangle = langle x_2, y rangle$。也就是说，$s(G) geqslant 2$，其中$s(G)$是$G$的传播。此外，如果$u(G)$表示$G$的更严格的一致扩展，则我们可以用$u(G) = 0$和$u(G)=1$完全表征有限群$G$。为了证明这些结果，我们首先建立了一个几乎简单群的约简。对于简单群，这个结果在2000年由Guralnick和Kantor用概率方法证明了，从那时起，几乎简单群就成为了几篇论文的主题。通过结合我们的约简定理和之前的工作，它仍然可以处理其群是李型例外群的群，这就是我们在本文中处理的情况。
{"title":"The spread of a finite group","authors":"Timothy C. Burness, R. Guralnick, Scott Harper","doi":"10.4007/ANNALS.2021.193.2.5","DOIUrl":"https://doi.org/10.4007/ANNALS.2021.193.2.5","url":null,"abstract":"A group $G$ is said to be $frac{3}{2}$-generated if every nontrivial element belongs to a generating pair. It is easy to see that if $G$ has this property then every proper quotient of $G$ is cyclic. In this paper we prove that the converse is true for finite groups, which settles a conjecture of Breuer, Guralnick and Kantor from 2008. In fact, we prove a much stronger result, which solves a problem posed by Brenner and Wiegold in 1975. Namely, if $G$ is a finite group and every proper quotient of $G$ is cyclic, then for any pair of nontrivial elements $x_1,x_2 in G$, there exists $y in G$ such that $G = langle x_1, y rangle = langle x_2, y rangle$. In other words, $s(G) geqslant 2$, where $s(G)$ is the spread of $G$. Moreover, if $u(G)$ denotes the more restrictive uniform spread of $G$, then we can completely characterise the finite groups $G$ with $u(G) = 0$ and $u(G)=1$. To prove these results, we first establish a reduction to almost simple groups. For simple groups, the result was proved by Guralnick and Kantor in 2000 using probabilistic methods and since then the almost simple groups have been the subject of several papers. By combining our reduction theorem and this earlier work, it remains to handle the groups whose socles are exceptional groups of Lie type and this is the case we treat in this paper.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87113868","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 : 2020-05-27DOI: 10.33048/SEMI.2020.17.052
A. Buturlakin, S. S. Presnyakov, D. Revin, S. A. Savin
Consider a triangle $ABC$ with given lengths $l_a,l_b,l_c$ of its internal angle bisectors. We prove that in general, it is impossible to construct a square of the same area as $ABC$ using a ruler and compass. Moreover, it is impossible to express the area of $ABC$ in radicals of $l_a,l_b,l_c$.
{"title":"Area of a triangle and angle bisectors","authors":"A. Buturlakin, S. S. Presnyakov, D. Revin, S. A. Savin","doi":"10.33048/SEMI.2020.17.052","DOIUrl":"https://doi.org/10.33048/SEMI.2020.17.052","url":null,"abstract":"Consider a triangle $ABC$ with given lengths $l_a,l_b,l_c$ of its internal angle bisectors. We prove that in general, it is impossible to construct a square of the same area as $ABC$ using a ruler and compass. Moreover, it is impossible to express the area of $ABC$ in radicals of $l_a,l_b,l_c$.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88499219","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 : 2020-05-24DOI: 10.1142/s0218216520420018
V. Bardakov, T. Kozlovskaya
In the present paper we study the singular pure braid group $SP_{n}$ for $n=2, 3$. We find generators, defining relations and the algebraical structure of these groups. In particular, we prove that $SP_{3}$ is a semi-direct product $SP_{3} = widetilde{V}_3 leftthreetimes mathbb{Z}$, where $widetilde{V}_3$ is an HNN-extension with base group $mathbb{Z}^2 * mathbb{Z}^2$ and cyclic associated subgroups. We prove that the center $Z(SP_3)$ of $SP_3$ is a direct factor in $SP_3$.
{"title":"On 3-strand singular pure braid group","authors":"V. Bardakov, T. Kozlovskaya","doi":"10.1142/s0218216520420018","DOIUrl":"https://doi.org/10.1142/s0218216520420018","url":null,"abstract":"In the present paper we study the singular pure braid group $SP_{n}$ for $n=2, 3$. We find generators, defining relations and the algebraical structure of these groups. In particular, we prove that $SP_{3}$ is a semi-direct product $SP_{3} = widetilde{V}_3 leftthreetimes mathbb{Z}$, where $widetilde{V}_3$ is an HNN-extension with base group $mathbb{Z}^2 * mathbb{Z}^2$ and cyclic associated subgroups. We prove that the center $Z(SP_3)$ of $SP_3$ is a direct factor in $SP_3$.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"22 3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86762135","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 $G$ be a finite primitive permutation group on a set $Omega$ with nontrivial point stabilizer $G_{alpha}$. We say that $G$ is extremely primitive if $G_{alpha}$ acts primitively on each of its orbits in $Omega setminus {alpha}$. These groups arise naturally in several different contexts and their study can be traced back to work of Manning in the 1920s. In this paper, we determine the almost simple extremely primitive groups with socle an exceptional group of Lie type. By combining this result with earlier work of Burness, Praeger and Seress, this completes the classification of the almost simple extremely primitive groups. Moreover, in view of results by Mann, Praeger and Seress, our main theorem gives a complete classification of all finite extremely primitive groups, up to finitely many affine exceptions (and it is conjectured that there are no exceptions). Along the way, we also establish several new results on base sizes for primitive actions of exceptional groups, which may be of independent interest.
{"title":"The Classification of Extremely Primitive Groups","authors":"Timothy C. Burness, Adam R. Thomas","doi":"10.1093/IMRN/RNAA369","DOIUrl":"https://doi.org/10.1093/IMRN/RNAA369","url":null,"abstract":"Let $G$ be a finite primitive permutation group on a set $Omega$ with nontrivial point stabilizer $G_{alpha}$. We say that $G$ is extremely primitive if $G_{alpha}$ acts primitively on each of its orbits in $Omega setminus {alpha}$. These groups arise naturally in several different contexts and their study can be traced back to work of Manning in the 1920s. In this paper, we determine the almost simple extremely primitive groups with socle an exceptional group of Lie type. By combining this result with earlier work of Burness, Praeger and Seress, this completes the classification of the almost simple extremely primitive groups. Moreover, in view of results by Mann, Praeger and Seress, our main theorem gives a complete classification of all finite extremely primitive groups, up to finitely many affine exceptions (and it is conjectured that there are no exceptions). Along the way, we also establish several new results on base sizes for primitive actions of exceptional groups, which may be of independent interest.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85488478","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 : 2020-05-21DOI: 10.33048/semi.2020.17.046
R. Bildanov, V. Goryachenko, A. Vasil’ev
A group $G$ is said to be factorized into subsets $A_1, A_2, ldots, A_ssubseteq G$ if every element $g$ in $G$ can be uniquely represented as $g=g_1g_2ldots g_s$, where $g_iin A_i$, $i=1,2,ldots,s$. We consider the following conjecture: for every finite group $G$ and every factorization $n=ab$ of its order, there is a factorization $G=AB$ with $|A|=a$ and $|B|=b$. We show that a minimal counterexample to this conjecture must be a nonabelian simple group and prove the conjecture for every finite group the nonabelian composition factors of which have orders less than $10,000$.
{"title":"Factoring nonabelian finite groups into two subsets","authors":"R. Bildanov, V. Goryachenko, A. Vasil’ev","doi":"10.33048/semi.2020.17.046","DOIUrl":"https://doi.org/10.33048/semi.2020.17.046","url":null,"abstract":"A group $G$ is said to be factorized into subsets $A_1, A_2, ldots, A_ssubseteq G$ if every element $g$ in $G$ can be uniquely represented as $g=g_1g_2ldots g_s$, where $g_iin A_i$, $i=1,2,ldots,s$. We consider the following conjecture: for every finite group $G$ and every factorization $n=ab$ of its order, there is a factorization $G=AB$ with $|A|=a$ and $|B|=b$. We show that a minimal counterexample to this conjecture must be a nonabelian simple group and prove the conjecture for every finite group the nonabelian composition factors of which have orders less than $10,000$.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"359 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75503836","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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