Pub Date : 2017-06-01DOI: 10.22108/IJGT.2017.11163
S. Kamornikov
Let $H$ be a prefrattini subgroup of a soluble finite group $G$. In the paper it is proved that there exist elements $x,y in G$ such that the equality $H cap H^x cap H^y = Phi (G)$ holds.
设$H$是可溶有限群$G$的prefrattini子群。本文证明了G$中存在元素$x,y，使得等式$H cap H^x cap H^y = φ (G)$成立。
{"title":"Intersections of prefrattini subgroups in finite soluble groups","authors":"S. Kamornikov","doi":"10.22108/IJGT.2017.11163","DOIUrl":"https://doi.org/10.22108/IJGT.2017.11163","url":null,"abstract":"Let $H$ be a prefrattini subgroup of a soluble finite group $G$. In the paper it is proved that there exist elements $x,y in G$ such that the equality $H cap H^x cap H^y = Phi (G)$ holds.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"6 1","pages":"1-5"},"PeriodicalIF":0.2,"publicationDate":"2017-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47613975","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 : 2017-04-08DOI: 10.22108/ijgt.2017.21612
T. Banakh, V. Gavrylkiv
A subset $B$ of a group $G$ is called a {em difference basis} of $G$ if each element $gin G$ can be written as the difference $g=ab^{-1}$ of some elements $a,bin B$. The smallest cardinality $|B|$ of a difference basis $Bsubset G$ is called the {em difference size} of $G$ and is denoted by $Delta[G]$. The fraction $eth[G]:=Delta[G]/{sqrt{|G|}}$ is called the {em difference characteristic} of $G$. We prove that for every $nin N$ the dihedral group $D_{2n}$ of order $2n$ has the difference characteristic $sqrt{2}leeth[D_{2n}]leqfrac{48}{sqrt{586}}approx1.983$. Moreover, if $nge 2cdot 10^{15}$, then $eth[D_{2n}]
{"title":"Difference bases in dihedral groups","authors":"T. Banakh, V. Gavrylkiv","doi":"10.22108/ijgt.2017.21612","DOIUrl":"https://doi.org/10.22108/ijgt.2017.21612","url":null,"abstract":"A subset $B$ of a group $G$ is called a {em difference basis} of $G$ if each element $gin G$ can be written as the difference $g=ab^{-1}$ of some elements $a,bin B$. The smallest cardinality $|B|$ of a difference basis $Bsubset G$ is called the {em difference size} of $G$ and is denoted by $Delta[G]$. The fraction $eth[G]:=Delta[G]/{sqrt{|G|}}$ is called the {em difference characteristic} of $G$. We prove that for every $nin N$ the dihedral group $D_{2n}$ of order $2n$ has the difference characteristic $sqrt{2}leeth[D_{2n}]leqfrac{48}{sqrt{586}}approx1.983$. Moreover, if $nge 2cdot 10^{15}$, then $eth[D_{2n}]<frac{4}{sqrt{6}}approx1.633$. Also we calculate the difference sizes and characteristics of all dihedral groups of cardinality $le80$.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"8 1","pages":"43-50"},"PeriodicalIF":0.2,"publicationDate":"2017-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41692253","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 : 2017-03-04DOI: 10.22108/IJGT.2017.21236
N. Ahanjideh
For a finite group $H$, let $cs(H)$ denote the set of non-trivial conjugacy class sizes of $H$ and $OC(H)$ be the set of the order components of $H$. In this paper, we show that if $S$ is a finite simple group with the disconnected prime graph and $G$ is a finite group such that $cs(S)=cs(G)$, then $|S|=|G/Z(G)|$ and $OC(S)=OC(G/Z(G))$. In particular, we show that for some finite simple group $S$, $G cong S times Z(G)$.
{"title":"Finite groups with the same conjugacy class sizes as a finite simple group","authors":"N. Ahanjideh","doi":"10.22108/IJGT.2017.21236","DOIUrl":"https://doi.org/10.22108/IJGT.2017.21236","url":null,"abstract":"For a finite group $H$, let $cs(H)$ denote the set of non-trivial conjugacy class sizes of $H$ and $OC(H)$ be the set of the order components of $H$. In this paper, we show that if $S$ is a finite simple group with the disconnected prime graph and $G$ is a finite group such that $cs(S)=cs(G)$, then $|S|=|G/Z(G)|$ and $OC(S)=OC(G/Z(G))$. In particular, we show that for some finite simple group $S$, $G cong S times Z(G)$.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"8 1","pages":"23-33"},"PeriodicalIF":0.2,"publicationDate":"2017-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42547662","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 : 2017-03-01DOI: 10.22108/IJGT.2017.11161
M. Akbari, A. Moghaddamfar
The noncommuting graph $nabla (G)$ of a group $G$ is a simple graph whose vertex set is the set of noncentral elements of $G$ and the edges of which are the ones connecting two noncommuting elements. We determine here, up to isomorphism, the structure of any finite nonabeilan group $G$ whose noncommuting graph is a split graph, that is, a graph whose vertex set can be partitioned into two sets such that the induced subgraph on one of them is a complete graph and the induced subgraph on the other is an independent set.
{"title":"Groups for which the noncommuting graph is a split graph","authors":"M. Akbari, A. Moghaddamfar","doi":"10.22108/IJGT.2017.11161","DOIUrl":"https://doi.org/10.22108/IJGT.2017.11161","url":null,"abstract":"The noncommuting graph $nabla (G)$ of a group $G$ is a simple graph whose vertex set is the set of noncentral elements of $G$ and the edges of which are the ones connecting two noncommuting elements. We determine here, up to isomorphism, the structure of any finite nonabeilan group $G$ whose noncommuting graph is a split graph, that is, a graph whose vertex set can be partitioned into two sets such that the induced subgraph on one of them is a complete graph and the induced subgraph on the other is an independent set.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"6 1","pages":"29-35"},"PeriodicalIF":0.2,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46258527","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 : 2017-03-01DOI: 10.22108/IJGT.2017.10890
G. Vincenzi
A subgroup X of a group G is said to be an H -subgroup if NG(X) X g X for each element g belonging to G. In (M. Bianchi and e.a., On nite soluble groups in which normality is a transitive relation, J. Group Theory, 3 (2000) 147{156.) the authors showed that nite groups in which every subgroup has the H -property are exactly soluble groups in which normality is a transitive relation. Here we extend this characterization to groups without simple sections.
在M. Bianchi和e.a. On n -可解群，其中每个子群都有H -性质，J.群论，3(2000)147{156.)中，作者证明了其中每个子群都具有H -性质的n -群是正态是可解群。在这里，我们将这个特征扩展到没有简单节的群。
{"title":"A characterization of soluble groups in which normality is a transitive relation","authors":"G. Vincenzi","doi":"10.22108/IJGT.2017.10890","DOIUrl":"https://doi.org/10.22108/IJGT.2017.10890","url":null,"abstract":"A subgroup X of a group G is said to be an H -subgroup if NG(X) X g X for each element g belonging to G. In (M. Bianchi and e.a., On nite soluble groups in which normality is a transitive relation, J. Group Theory, 3 (2000) 147{156.) the authors showed that nite groups in which every subgroup has the H -property are exactly soluble groups in which normality is a transitive relation. Here we extend this characterization to groups without simple sections.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"6 1","pages":"21-27"},"PeriodicalIF":0.2,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48712130","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 : 2017-03-01DOI: 10.22108/IJGT.2017.10506
L. Poinsot
This contribution mainly focuses on some aspects of Lipschitz groups, i.e., metrizable groups with Lipschitz multiplication and inversion map. In the main result it is proved that metric groups, with a translation-invariant metric, may be characterized as particular group objects in the category of metric spaces and Lipschitz maps. Moreover, up to an adjustment of the metric, any metrizable abelian group also is shown to be a Lipschitz group. Finally we present a result similar to the fact that any topological nilpotent element x in a Banach algebra gives rise to an invertible element 1 x, in the setting of complete Lipschitz groups.
{"title":"LIPSCHITZ GROUPS AND LIPSCHITZ MAPS","authors":"L. Poinsot","doi":"10.22108/IJGT.2017.10506","DOIUrl":"https://doi.org/10.22108/IJGT.2017.10506","url":null,"abstract":"This contribution mainly focuses on some aspects of Lipschitz groups, i.e., metrizable groups with Lipschitz multiplication and inversion map. In the main result it is proved that metric groups, with a translation-invariant metric, may be characterized as particular group objects in the category of metric spaces and Lipschitz maps. Moreover, up to an adjustment of the metric, any metrizable abelian group also is shown to be a Lipschitz group. Finally we present a result similar to the fact that any topological nilpotent element x in a Banach algebra gives rise to an invertible element 1 x, in the setting of complete Lipschitz groups.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"6 1","pages":"9-16"},"PeriodicalIF":0.2,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46806047","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 concept of the bipartite divisor graph for integer subsets has been considered in [M. A. Iranmanesh and C. E. Praeger, Bipartite divisor graphs for integer subsets, Graphs Combin., 26 (2010) 95--105.]. In this paper, we will consider this graph for the set of character degrees of a finite group $G$ and obtain some properties of this graph. We show that if $G$ is a solvable group, then the number of connected components of this graph is at most $2$ and if $G$ is a non-solvable group, then it has at most $3$ connected components. We also show that the diameter of a connected bipartite divisor graph is bounded by $7$ and obtain some properties of groups whose graphs are complete bipartite graphs.
{"title":"On bipartite divisor graph for character degrees","authors":"S. A. Moosavi","doi":"10.22108/IJGT.2017.9852","DOIUrl":"https://doi.org/10.22108/IJGT.2017.9852","url":null,"abstract":"The concept of the bipartite divisor graph for integer subsets has been considered in [M. A. Iranmanesh and C. E. Praeger, Bipartite divisor graphs for integer subsets, Graphs Combin., 26 (2010) 95--105.]. In this paper, we will consider this graph for the set of character degrees of a finite group $G$ and obtain some properties of this graph. We show that if $G$ is a solvable group, then the number of connected components of this graph is at most $2$ and if $G$ is a non-solvable group, then it has at most $3$ connected components. We also show that the diameter of a connected bipartite divisor graph is bounded by $7$ and obtain some properties of groups whose graphs are complete bipartite graphs.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"6 1","pages":"1-7"},"PeriodicalIF":0.2,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43729574","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 : 2017-02-15DOI: 10.22108/IJGT.2017.21215
M. Pellegrini
In this paper we highlight the connection between certain classes of regular subgroups of the affine group $AGL_n(F)$, $F$ a field, and associative nilpotent $F$-algebras of dimension $n$. We also describe how the classification of projective congruence classes of square matrices is equivalent to the classification of regular subgroups of particular shape.
{"title":"Regular subgroups, nilpotent algebras and projectively congruent matrices","authors":"M. Pellegrini","doi":"10.22108/IJGT.2017.21215","DOIUrl":"https://doi.org/10.22108/IJGT.2017.21215","url":null,"abstract":"In this paper we highlight the connection between certain classes of regular subgroups of the affine group $AGL_n(F)$, $F$ a field, and associative nilpotent $F$-algebras of dimension $n$. We also describe how the classification of projective congruence classes of square matrices is equivalent to the classification of regular subgroups of particular shape.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"7 1","pages":"51-56"},"PeriodicalIF":0.2,"publicationDate":"2017-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48010349","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 : 2017-01-17DOI: 10.22108/IJGT.2017.21216
A. Beltrán, M. J. Felipe, C. Melchor
Some of the results of this paper are part of the third author's Ph.D. thesis at the University Jaume I of Castellon, who is financially supported by a predoctoral grant of this university. The first and second authors are supported by the Valencian Government, Proyecto PROMETEOII/2015/011. The first and the third authors are also partially supported by Universitat Jaume I, grant P11B2015-77.
{"title":"Conjugacy classes contained in normal subgroups: an overview","authors":"A. Beltrán, M. J. Felipe, C. Melchor","doi":"10.22108/IJGT.2017.21216","DOIUrl":"https://doi.org/10.22108/IJGT.2017.21216","url":null,"abstract":"Some of the results of this paper are part of the third author's Ph.D. thesis at the University Jaume \u0000I of Castellon, who is financially supported by a predoctoral grant of this university. The first and \u0000second authors are supported by the Valencian Government, Proyecto PROMETEOII/2015/011. The \u0000 first and the third authors are also partially supported by Universitat Jaume I, grant P11B2015-77.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"7 1","pages":"23-36"},"PeriodicalIF":0.2,"publicationDate":"2017-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42879452","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 : 2017-01-07DOI: 10.22108/IJGT.2017.21219
R. Soleimani
Let $G$ be a group and $Aut^{Phi}(G)$ denote the group of all automorphisms of $G$ centralizing $G/Phi(G)$ elementwise. In this paper, we characterize the finite $p$-groups $G$ with cyclic Frattini subgroup for which $|Aut^{Phi}(G):Inn(G)|=p$.
{"title":"Automorphisms of a finite $p$-group with cyclic Frattini subgroup","authors":"R. Soleimani","doi":"10.22108/IJGT.2017.21219","DOIUrl":"https://doi.org/10.22108/IJGT.2017.21219","url":null,"abstract":"Let $G$ be a group and $Aut^{Phi}(G)$ denote the group of all automorphisms of $G$ centralizing $G/Phi(G)$ elementwise. In this paper, we characterize the finite $p$-groups $G$ with cyclic Frattini subgroup for which $|Aut^{Phi}(G):Inn(G)|=p$.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"7 1","pages":"9-16"},"PeriodicalIF":0.2,"publicationDate":"2017-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41928797","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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