Let $G$ be a topological group and let $K,Lsubseteq G$ be closed subgroups, with $Ksubseteq L$. We prove that if $L$ is a locally compact pro-Lie group, then the map $q:G/Kto G/L$ is a fibration. As an application of this, we obtain two older results by Skljarenko, Madison and Mostert.
{"title":"Fibrations and coset spaces for locally compact groups","authors":"Linus Kramer, Raquel Murat García","doi":"arxiv-2408.03843","DOIUrl":"https://doi.org/arxiv-2408.03843","url":null,"abstract":"Let $G$ be a topological group and let $K,Lsubseteq G$ be closed subgroups,\u0000with $Ksubseteq L$. We prove that if $L$ is a locally compact pro-Lie group, then the map\u0000$q:G/Kto G/L$ is a fibration. As an application of this, we obtain two older\u0000results by Skljarenko, Madison and Mostert.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934016","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 this paper we present some geometrical representations of the Frobenius group of order $21$ (henceforth, $F_{21}$). The main focus is on investigating the group of common automorphisms of two orthogonal Fano planes and the automorphism group of a suitably oriented Fano plane. We show that both groups are isomorphic to $F_{21},$ independently of the choice of the two orthogonal Fano planes and of the choice of the orientation. We show, moreover, that any triangular embedding of the complete graph $K_7$ into a surface is isomorphic to the classical toroidal biembedding and hence is face $2$-colorable, with the two color classes defining a pair of orthogonal Fano planes. As a consequence, we show that, for any triangular embedding of $K_7$ into a surface, the group of the automorphisms that preserve the color classes is the Frobenius group of order $21.$ This way we provide three geometrical representations of $F_{21}$. Also, we apply the representation in terms of two orthogonal Fano planes to give an alternative proof that $F_{21}$ is the automorphism group of the Kirkman triple system of order $15$ that is usually denoted as #61.
{"title":"Orthogonal and oriented Fano planes, triangular embeddings of $K_7,$ and geometrical representations of the Frobenius group $F_{21}$","authors":"Simone Costa, Marco Pavone","doi":"arxiv-2408.03743","DOIUrl":"https://doi.org/arxiv-2408.03743","url":null,"abstract":"In this paper we present some geometrical representations of the Frobenius\u0000group of order $21$ (henceforth, $F_{21}$). The main focus is on investigating\u0000the group of common automorphisms of two orthogonal Fano planes and the\u0000automorphism group of a suitably oriented Fano plane. We show that both groups\u0000are isomorphic to $F_{21},$ independently of the choice of the two orthogonal\u0000Fano planes and of the choice of the orientation. We show, moreover, that any triangular embedding of the complete graph $K_7$\u0000into a surface is isomorphic to the classical toroidal biembedding and hence is\u0000face $2$-colorable, with the two color classes defining a pair of orthogonal\u0000Fano planes. As a consequence, we show that, for any triangular embedding of\u0000$K_7$ into a surface, the group of the automorphisms that preserve the color\u0000classes is the Frobenius group of order $21.$ This way we provide three geometrical representations of $F_{21}$. Also, we\u0000apply the representation in terms of two orthogonal Fano planes to give an\u0000alternative proof that $F_{21}$ is the automorphism group of the Kirkman triple\u0000system of order $15$ that is usually denoted as #61.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934015","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}
Peter J. Cameron, Rishabh Chakraborty, Rajat Kanti Nath, Deiborlang Nongsiang
Let $G$ be a group. Associate a graph $mathcal{E}_G$ (called the co-Engel graph of $G$) with $G$ whose vertex set is $G$ and two distinct vertices $x$ and $y$ are adjacent if $[x, {}_k y] neq 1$ and $[y, {}_k x] neq 1$ for all positive integer $k$. This graph, under the name ``Engel graph'', was introduced by Abdollahi. Let $L(G)$ be the set of all left Engel elements of $G$. In this paper, we realize the induced subgraph of co-Engel graphs of certain finite non-Engel groups $G$ induced by $G setminus L(G)$. We write $mathcal{E}^-(G)$ to denote the subgraph of $mathcal{E}_G$ induced by $G setminus L(G)$. We also compute genus, various spectra, energies and Zagreb indices of $mathcal{E}^-(G)$ for those groups. As a consequence, we determine (up to isomorphism) all finite non-Engel group $G$ such that the clique number is at most $4$ and $mathcal{E}^-$ is toroidal or projective. Further, we show that $coeng{G}$ is super integral and satisfies the E-LE conjecture and the Hansen--Vuki{v{c}}evi{'c} conjecture for the groups considered in this paper.
{"title":"Co-Engel graphs of certain finite non-Engel groups","authors":"Peter J. Cameron, Rishabh Chakraborty, Rajat Kanti Nath, Deiborlang Nongsiang","doi":"arxiv-2408.03879","DOIUrl":"https://doi.org/arxiv-2408.03879","url":null,"abstract":"Let $G$ be a group. Associate a graph $mathcal{E}_G$ (called the co-Engel\u0000graph of $G$) with $G$ whose vertex set is $G$ and two distinct vertices $x$\u0000and $y$ are adjacent if $[x, {}_k y] neq 1$ and $[y, {}_k x] neq 1$ for all\u0000positive integer $k$. This graph, under the name ``Engel graph'', was\u0000introduced by Abdollahi. Let $L(G)$ be the set of all left Engel elements of\u0000$G$. In this paper, we realize the induced subgraph of co-Engel graphs of\u0000certain finite non-Engel groups $G$ induced by $G setminus L(G)$. We write\u0000$mathcal{E}^-(G)$ to denote the subgraph of $mathcal{E}_G$ induced by $G\u0000setminus L(G)$. We also compute genus, various spectra, energies and Zagreb\u0000indices of $mathcal{E}^-(G)$ for those groups. As a consequence, we determine\u0000(up to isomorphism) all finite non-Engel group $G$ such that the clique number\u0000is at most $4$ and $mathcal{E}^-$ is toroidal or projective. Further, we show\u0000that $coeng{G}$ is super integral and satisfies the E-LE conjecture and the\u0000Hansen--Vuki{v{c}}evi{'c} conjecture for the groups considered in this paper.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934013","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}
A positive integer $m$ is called a Hall number if any finite group of order precisely divisible by $m$ has a Hall subgroup of order $m$. We prove that, except for the obvious examples, the three integers $12$, $24$ and $60$ are the only Hall numbers, solving a problem proposed by Jiping Zhang.
{"title":"The exceptional Hall numbers","authors":"Zheng Guo, Yong Hu, Cai Heng Li","doi":"arxiv-2408.03184","DOIUrl":"https://doi.org/arxiv-2408.03184","url":null,"abstract":"A positive integer $m$ is called a Hall number if any finite group of order\u0000precisely divisible by $m$ has a Hall subgroup of order $m$. We prove that,\u0000except for the obvious examples, the three integers $12$, $24$ and $60$ are the\u0000only Hall numbers, solving a problem proposed by Jiping Zhang.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934018","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 this paper it is shown that for the natural density (among the positive integers) of the orders of the finite quotients of every ordinary triangle group is zero, using a modification of a component of a 1976 theorem of Bertram on large cyclic subgroups of finite groups, and the Turan-Kubilius inequality from asymptotic number theory. This answers a challenging question raised by Tucker, based on some work for special cases by May and Zimmerman and himself.
{"title":"The density of orders of quotients of triangle groups","authors":"Darius Young","doi":"arxiv-2408.02264","DOIUrl":"https://doi.org/arxiv-2408.02264","url":null,"abstract":"In this paper it is shown that for the natural density (among the positive\u0000integers) of the orders of the finite quotients of every ordinary triangle\u0000group is zero, using a modification of a component of a 1976 theorem of Bertram\u0000on large cyclic subgroups of finite groups, and the Turan-Kubilius inequality\u0000from asymptotic number theory. This answers a challenging question raised by\u0000Tucker, based on some work for special cases by May and Zimmerman and himself.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934017","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}
María José Felipe, Marc Kelly Jean-Philippe, Víctor Sotomayor
Let $p$ be a prime. In this paper we classify the $p$-structure of those finite $p$-separable groups such that, given any three non-central conjugacy classes of $p$-regular elements, two of them necessarily have coprime lengths.
{"title":"Classification of groups whose common divisor graph on $p$-regular classes has no triangles","authors":"María José Felipe, Marc Kelly Jean-Philippe, Víctor Sotomayor","doi":"arxiv-2408.02818","DOIUrl":"https://doi.org/arxiv-2408.02818","url":null,"abstract":"Let $p$ be a prime. In this paper we classify the $p$-structure of those\u0000finite $p$-separable groups such that, given any three non-central conjugacy\u0000classes of $p$-regular elements, two of them necessarily have coprime lengths.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934008","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}
The article deals with finite groups in which commutators have prime power order (CPPO-groups). We show that if G is a soluble CPPO-group, then the order of the commutator subgroup G' is divisible by at most two primes.
文章涉及换元具有素幂阶的有限群（CPPO-群）。我们证明，如果 G 是一个可溶的 CPPO 群，那么换元子群 G' 的阶最多可被两个素数整除。
{"title":"On soluble groups in which commutators have prime power order","authors":"Mateus Figueiredo, Pavel Shumyatsky","doi":"arxiv-2408.01974","DOIUrl":"https://doi.org/arxiv-2408.01974","url":null,"abstract":"The article deals with finite groups in which commutators have prime power\u0000order (CPPO-groups). We show that if G is a soluble CPPO-group, then the order\u0000of the commutator subgroup G' is divisible by at most two primes.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141968825","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 $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms. We provide a complete classification of a finite group $G$ in which every maximal $A$-invariant subgroup containing the normalizer of some $A$-invariant Sylow subgroup is nilpotent. Moreover, we show that both the hypothesis that every maximal $A$-invariant subgroup of $G$ containing the normalizer of some $A$-invariant Sylow subgroup is nilpotent and the hypothesis that every non-nilpotent maximal $A$-invariant subgroup of $G$ is normal are equivalent.
{"title":"Finite groups with some particular maximal invariant subgroups being nilpotent or all non-nilpotent maximal invariant subgroups being normal","authors":"Jiangtao Shi, Fanjie Xu","doi":"arxiv-2408.01249","DOIUrl":"https://doi.org/arxiv-2408.01249","url":null,"abstract":"Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by\u0000automorphisms. We provide a complete classification of a finite group $G$ in\u0000which every maximal $A$-invariant subgroup containing the normalizer of some\u0000$A$-invariant Sylow subgroup is nilpotent. Moreover, we show that both the\u0000hypothesis that every maximal $A$-invariant subgroup of $G$ containing the\u0000normalizer of some $A$-invariant Sylow subgroup is nilpotent and the hypothesis\u0000that every non-nilpotent maximal $A$-invariant subgroup of $G$ is normal are\u0000equivalent.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934007","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}
The ideal membership problem asks whether an element in the ring belongs to the given ideal. In this paper, we propose a function that reflecting the complexity of the ideal membership problem in the ring of Laurent polynomials with integer coefficients. We also connect the complexity function we define to the Dehn function of a metabelian group, in the hope of constructing a metabelian group with superexponential Dehn function.
{"title":"From Ideal Membership Problem for polynomial rings to Dehn Functions of Metabelian Groups","authors":"Wenhao Wang","doi":"arxiv-2408.01518","DOIUrl":"https://doi.org/arxiv-2408.01518","url":null,"abstract":"The ideal membership problem asks whether an element in the ring belongs to\u0000the given ideal. In this paper, we propose a function that reflecting the\u0000complexity of the ideal membership problem in the ring of Laurent polynomials\u0000with integer coefficients. We also connect the complexity function we define to\u0000the Dehn function of a metabelian group, in the hope of constructing a\u0000metabelian group with superexponential Dehn function.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934019","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}
Andrey R. Chekhlov, Peter V. Danchev, Brendan Goldsmith, Patrick W. Keef
This paper targets to generalize the notion of Hopfian groups in the commutative case by defining the so-called {bf relatively Hopfian groups} and {bf weakly Hopfian groups}, and establishing some their crucial properties and characterizations. Specifically, we prove that for a reduced Abelian $p$-group $G$ such that $p^{omega}G$ is Hopfian (in particular, is finite), the notions of relative Hopficity and ordinary Hopficity do coincide. We also show that if $G$ is a reduced Abelian $p$-group such that $p^{omega}G$ is bounded and $G/p^{omega}G$ is Hopfian, then $G$ is relatively Hopfian. This allows us to construct a reduced relatively Hopfian Abelian $p$-group $G$ with $p^{omega}G$ an infinite elementary group such that $G$ is {bf not} Hopfian. In contrast, for reduced torsion-free groups, we establish that the relative and ordinary Hopficity are equivalent. Moreover, the mixed case is explored as well, showing that the structure of both relatively and weakly Hopfian groups can be quite complicated.
{"title":"Two Generalizations of Hopfian Abelian Groupa","authors":"Andrey R. Chekhlov, Peter V. Danchev, Brendan Goldsmith, Patrick W. Keef","doi":"arxiv-2408.01277","DOIUrl":"https://doi.org/arxiv-2408.01277","url":null,"abstract":"This paper targets to generalize the notion of Hopfian groups in the\u0000commutative case by defining the so-called {bf relatively Hopfian groups} and\u0000{bf weakly Hopfian groups}, and establishing some their crucial properties and\u0000characterizations. Specifically, we prove that for a reduced Abelian $p$-group\u0000$G$ such that $p^{omega}G$ is Hopfian (in particular, is finite), the notions\u0000of relative Hopficity and ordinary Hopficity do coincide. We also show that if\u0000$G$ is a reduced Abelian $p$-group such that $p^{omega}G$ is bounded and\u0000$G/p^{omega}G$ is Hopfian, then $G$ is relatively Hopfian. This allows us to\u0000construct a reduced relatively Hopfian Abelian $p$-group $G$ with $p^{omega}G$\u0000an infinite elementary group such that $G$ is {bf not} Hopfian. In contrast,\u0000for reduced torsion-free groups, we establish that the relative and ordinary\u0000Hopficity are equivalent. Moreover, the mixed case is explored as well, showing\u0000that the structure of both relatively and weakly Hopfian groups can be quite\u0000complicated.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934021","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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