Pub Date : 2020-12-01DOI: 10.22108/IJGT.2019.119581.1577
E. Detomi, M. Morigi
Let $gamma_n=[x_1,ldots,x_n]$ be the $n$th lower central word. Suppose that $G$ is a profinite group where the conjugacy classes $x^{gamma_n(G)}$ contains less than $2^{aleph_0}$ elements for any $x in G$. We prove that then $gamma_{n+1}(G)$ has finite order. This generalizes the much celebrated theorem of B. H. Neumann that says that the commutator subgroup of a BFC-group is finite. Moreover, it implies that a profinite group $G$ is finite-by-nilpotent if and only if there is a positive integer $n$ such that $x^{gamma_n(G)}$ contains less than $2^{aleph_0}$ elements, for any $xin G$.
{"title":"On finite-by-nilpotent profinite groups","authors":"E. Detomi, M. Morigi","doi":"10.22108/IJGT.2019.119581.1577","DOIUrl":"https://doi.org/10.22108/IJGT.2019.119581.1577","url":null,"abstract":"Let $gamma_n=[x_1,ldots,x_n]$ be the $n$th lower central word. Suppose that $G$ is a profinite group where the conjugacy classes $x^{gamma_n(G)}$ contains less than $2^{aleph_0}$ elements for any $x in G$. We prove that then $gamma_{n+1}(G)$ has finite order. This generalizes the much celebrated theorem of B. H. Neumann that says that the commutator subgroup of a BFC-group is finite. Moreover, it implies that a profinite group $G$ is finite-by-nilpotent if and only if there is a positive integer $n$ such that $x^{gamma_n(G)}$ contains less than $2^{aleph_0}$ elements, for any $xin G$.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"9 1","pages":"223-229"},"PeriodicalIF":0.2,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41609594","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 : 2020-09-01DOI: 10.22108/IJGT.2018.112602.1500
V. Monakhov, I. Sokhor
Let $G$ be a finite group in which every Sylow subgroup is seminormal or abnormal. We prove that $G$ has a Sylow tower. We establish that if a group has a maximal subgroup with Sylow subgroups under the same conditions, then this group is soluble.
{"title":"Finite groups with seminormal or abnormal Sylow subgroups","authors":"V. Monakhov, I. Sokhor","doi":"10.22108/IJGT.2018.112602.1500","DOIUrl":"https://doi.org/10.22108/IJGT.2018.112602.1500","url":null,"abstract":"Let $G$ be a finite group in which every Sylow subgroup is seminormal or abnormal. We prove that $G$ has a Sylow tower. We establish that if a group has a maximal subgroup with Sylow subgroups under the same conditions, then this group is soluble.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"9 1","pages":"139-142"},"PeriodicalIF":0.2,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46675371","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 : 2020-06-01DOI: 10.22108/IJGT.2019.114770.1522
Á. Figula, A. Al-Abayechi
The main result of our consideration is the proof of the centrally nilpotency of class two property for connected topological proper loops $L$ of dimension $le 3$ which have an at most six-dimensional solvable indecomposable Lie group as their multiplication group. This theorem is obtained from our previous classification by the investigation of six-dimensional indecomposable solvable multiplication Lie groups having a five-dimensional nilradical. We determine the Lie algebras of these multiplication groups and the subalgebras of the corresponding inner mapping groups.
{"title":"Topological loops with solvable multiplication groups of dimension at most six are centrally nilpotent","authors":"Á. Figula, A. Al-Abayechi","doi":"10.22108/IJGT.2019.114770.1522","DOIUrl":"https://doi.org/10.22108/IJGT.2019.114770.1522","url":null,"abstract":"The main result of our consideration is the proof of the centrally nilpotency of class two property for connected topological proper loops $L$ of dimension $le 3$ which have an at most six-dimensional solvable indecomposable Lie group as their multiplication group. This theorem is obtained from our previous classification by the investigation of six-dimensional indecomposable solvable multiplication Lie groups having a five-dimensional nilradical. We determine the Lie algebras of these multiplication groups and the subalgebras of the corresponding inner mapping groups.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"9 1","pages":"81-94"},"PeriodicalIF":0.2,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49016603","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 : 2020-04-17DOI: 10.22108/IJGT.2020.122081.1604
F. Russo, S. Muhie
A finite group $G$, in which two randomly chosen subgroups $H$ and $K$ commute, has been classified by Iwasawa in 1941. It is possible to define a probabilistic notion, which ``measures the distance'' of $G$ from the groups of Iwasawa. Here we introduce the generalized subgroup commutativity degree $gsd(G)$ for two arbitrary sublattices $mathrm{S}(G)$ and $mathrm{T}(G)$ of the lattice of subgroups $mathrm{L}(G)$ of $G$. Upper and lower bounds for $gsd(G)$ are shown and we study the behaviour of $gsd(G)$ with respect to subgroups and quotients, showing new numerical restrictions.
{"title":"The probability of commuting subgroups in arbitrary lattices of subgroups","authors":"F. Russo, S. Muhie","doi":"10.22108/IJGT.2020.122081.1604","DOIUrl":"https://doi.org/10.22108/IJGT.2020.122081.1604","url":null,"abstract":"A finite group $G$, in which two randomly chosen subgroups $H$ and $K$ commute, has been classified by Iwasawa in 1941. It is possible to define a probabilistic notion, which ``measures the distance'' of $G$ from the groups of Iwasawa. Here we introduce the generalized subgroup commutativity degree $gsd(G)$ for two arbitrary sublattices $mathrm{S}(G)$ and $mathrm{T}(G)$ of the lattice of subgroups $mathrm{L}(G)$ of $G$. Upper and lower bounds for $gsd(G)$ are shown and we study the behaviour of $gsd(G)$ with respect to subgroups and quotients, showing new numerical restrictions.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44273650","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 : 2020-03-25DOI: 10.22108/IJGT.2020.121125.1595
P. Longobardi, M. Maj
We study some inverse problems of small doubling type in the class of $m$-Engel groups. In particular we investigate the structure of a finite subset $S$ of a torsion-free $m$-Engel group if $|S^2| = 2|S|+b$, where $0 leq b leq |S|-4$, for some values of $b$.
研究了$m$-Engel群类的小倍型反问题。特别地，我们研究了无挠$m$-Engel群的有限子集$S$的结构，当$|S^2| = 2|S|+b$，其中$0 leq b leq |S|-4$，对于$b$的某些值。
{"title":"Small doubling in $m$-Engel groups","authors":"P. Longobardi, M. Maj","doi":"10.22108/IJGT.2020.121125.1595","DOIUrl":"https://doi.org/10.22108/IJGT.2020.121125.1595","url":null,"abstract":"We study some inverse problems of small doubling type in the class of $m$-Engel groups. In particular we investigate the structure of a finite subset $S$ of a torsion-free $m$-Engel group if $|S^2| = 2|S|+b$, where $0 leq b leq |S|-4$, for some values of $b$.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43932165","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 : 2020-03-25DOI: 10.22108/IJGT.2020.120552.1588
M. Pourhasan, H. Doostie
For a given odd prime $p$, we investigate the power graphs of three classes of finite groups: the elementary abelian groups of exponent $p$, and the extra special groups of exponents $p$ or $p^2$. We show that these power graphs are Eulerian for every $p$. As a corollary, we describe two classes of non-isomorphic groups with isomorphic power graphs. In addition, we prove that the clique graphs of the power graphs of two considered classes are complete.
{"title":"On the power graphs of elementary abelian and extra special $p$-groups","authors":"M. Pourhasan, H. Doostie","doi":"10.22108/IJGT.2020.120552.1588","DOIUrl":"https://doi.org/10.22108/IJGT.2020.120552.1588","url":null,"abstract":"For a given odd prime $p$, we investigate the power graphs of three classes of finite groups: the elementary abelian groups of exponent $p$, and the extra special groups of exponents $p$ or $p^2$. We show that these power graphs are Eulerian for every $p$. As a corollary, we describe two classes of non-isomorphic groups with isomorphic power graphs. In addition, we prove that the clique graphs of the power graphs of two considered classes are complete.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49314703","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 : 2020-03-06DOI: 10.22108/IJGT.2020.119054.1569
D. Robinson
The effect of restricting the set of primes dividing the orders of the finite quotients of a group is investigated. Particular attention is paid to abelian, soluble, locally soluble and locally finite groups. The connection with the extraction of roots is explored.
{"title":"On Infinite Groups whose Finite Quotients have Restricted Prime Divisors","authors":"D. Robinson","doi":"10.22108/IJGT.2020.119054.1569","DOIUrl":"https://doi.org/10.22108/IJGT.2020.119054.1569","url":null,"abstract":"The effect of restricting the set of primes dividing the orders of the finite quotients of a group is investigated. Particular attention is paid to abelian, soluble, locally soluble and locally finite groups. The connection with the extraction of roots is explored.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43814210","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 : 2020-03-01DOI: 10.22108/IJGT.2018.112531.1496
A. Lucchini, Mariapia Moscatiello
For a finite group group, denote by $mathcal V(G)$ the smallest positive integer $k$ with the property that the probability of generating $G$ by $k$ randomly chosen elements is at least $1/e.$ Let $G$ be a finite soluble group. {Assume} that for every $pin pi(G)$ there exists $G_pleq G$ such that $p$ does not divide $|G:G_p|$ and ${mathcal V}(G_p)leq d.$ Then ${mathcal V}(G)leq d+7.$
{"title":"A probabilistic version of a theorem of lászló kovács and hyo-seob sim","authors":"A. Lucchini, Mariapia Moscatiello","doi":"10.22108/IJGT.2018.112531.1496","DOIUrl":"https://doi.org/10.22108/IJGT.2018.112531.1496","url":null,"abstract":"For a finite group group, denote by $mathcal V(G)$ the smallest positive integer $k$ with the property that the probability of generating $G$ by $k$ randomly chosen elements is at least $1/e.$ Let $G$ be a finite soluble group. {Assume} that for every $pin pi(G)$ there exists $G_pleq G$ such that $p$ does not divide $|G:G_p|$ and ${mathcal V}(G_p)leq d.$ Then ${mathcal V}(G)leq d+7.$","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"9 1","pages":"1-6"},"PeriodicalIF":0.2,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45125412","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 : 2020-03-01DOI: 10.22108/IJGT.2019.114315.1518
C. Delizia, Chiara Nicotera
The purpose of this paper is to present a comprehensive overview of known and new results concerning the structure of groups in which all subgroups, except those having a given property, are either self-centralizing or self-normalizing.
{"title":"Groups with many self-centralizing or self-normalizing subgroups","authors":"C. Delizia, Chiara Nicotera","doi":"10.22108/IJGT.2019.114315.1518","DOIUrl":"https://doi.org/10.22108/IJGT.2019.114315.1518","url":null,"abstract":"The purpose of this paper is to present a comprehensive overview of known and new results concerning the structure of groups in which all subgroups, except those having a given property, are either self-centralizing or self-normalizing.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"9 1","pages":"43-57"},"PeriodicalIF":0.2,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42642971","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 : 2020-03-01DOI: 10.22108/IJGT.2019.114469.1521
Adolfo Ballester Bolinches, R. Esteban-Romero, P. Jiménez-Seral, H. Meng
In this survey we present some significant bounds for the number of maximal subgroups of a given index of a finite group. As a consequence, new bounds for the number of random generators needed to generate a finite $d$-generated group with high probability which are significantly tighter than the ones obtained in the paper of Jaikin-Zapirain and Pyber (Random generation of finite and profinite groups and group enumeration, emph{Ann. Math.}, textbf{183} (2011) 769--814) are obtained. The results of Jaikin-Zapirain and Pyber, as well as other results of Lubotzky, Detomi, and Lucchini, appear as particular cases of our theorems.
{"title":"The number of maximal subgroups and probabilistic generation of finite groups","authors":"Adolfo Ballester Bolinches, R. Esteban-Romero, P. Jiménez-Seral, H. Meng","doi":"10.22108/IJGT.2019.114469.1521","DOIUrl":"https://doi.org/10.22108/IJGT.2019.114469.1521","url":null,"abstract":"In this survey we present some significant bounds for the number of maximal subgroups of a given index of a finite group. As a consequence, new bounds for the number of random generators needed to generate a finite $d$-generated group with high probability which are significantly tighter than the ones obtained in the paper of Jaikin-Zapirain and Pyber (Random generation of finite and profinite groups and group enumeration, emph{Ann. Math.}, textbf{183} (2011) 769--814) are obtained. The results of Jaikin-Zapirain and Pyber, as well as other results of Lubotzky, Detomi, and Lucchini, appear as particular cases of our theorems.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"9 1","pages":"31-42"},"PeriodicalIF":0.2,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47387516","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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