Pub Date : 2016-12-20DOI: 10.22108/IJGT.2017.21674
M. Dixon, L. A. Kurdachenko, I. Subbotin
This paper deals with the mutual relationships between the factor group G/ζ(G) (respectively G/ζk(G)) and G ′ (respectively γk+1(G) and G ). It is proved that if G/ζ(G) (respectively G/ζk(G)) has finite 0-rank, then G ′ (respectively γk+1(G) and G ) also have finite 0-rank. Furthermore, bounds for the 0-ranks of G′, γk+1(G) and G N are obtained.
{"title":"On the relationships between the factors of the upper and lower central series in some non-periodic groups","authors":"M. Dixon, L. A. Kurdachenko, I. Subbotin","doi":"10.22108/IJGT.2017.21674","DOIUrl":"https://doi.org/10.22108/IJGT.2017.21674","url":null,"abstract":"This paper deals with the mutual relationships between the factor group G/ζ(G) (respectively G/ζk(G)) and G ′ (respectively γk+1(G) and G ). It is proved that if G/ζ(G) (respectively G/ζk(G)) has finite 0-rank, then G ′ (respectively γk+1(G) and G ) also have finite 0-rank. Furthermore, bounds for the 0-ranks of G′, γk+1(G) and G N are obtained.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"7 1","pages":"37-50"},"PeriodicalIF":0.2,"publicationDate":"2016-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68205735","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 generalize some transfer theorems. In particular, we derive one of the main results of Gagola (Contemp Math., 524 (2010) 49-60) from our results.
{"title":"A note on transfer theorems","authors":"Haoran Yu","doi":"10.22108/IJGT.2016.9851","DOIUrl":"https://doi.org/10.22108/IJGT.2016.9851","url":null,"abstract":"In this paper, we generalize some transfer theorems. In particular, we derive one of the main results of Gagola (Contemp Math., 524 (2010) 49-60) from our results.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"5 1","pages":"47-52"},"PeriodicalIF":0.2,"publicationDate":"2016-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68205546","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 study the coprime graph of a group G. The coprime graph of a group G, denoted by G, is a graph whose vertices are elements of G and two distinct vertices x and y are adjacent if and only if ( jxj;jyj) = 1. In this paper, we show that ( G) = !( G) : We classify all the groups which G is a complete r partite graph or a planar graph. Also we study the automorphism group of G.
{"title":"A NOTE ON THE COPRIME GRAPH OF A GROUP","authors":"H. Dorbidi","doi":"10.22108/IJGT.2016.9125","DOIUrl":"https://doi.org/10.22108/IJGT.2016.9125","url":null,"abstract":"In this paper we study the coprime graph of a group G. The coprime graph of a group G, denoted by G, is a graph whose vertices are elements of G and two distinct vertices x and y are adjacent if and only if ( jxj;jyj) = 1. In this paper, we show that ( G) = !( G) : We classify all the groups which G is a complete r partite graph or a planar graph. Also we study the automorphism group of G.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"5 1","pages":"17-22"},"PeriodicalIF":0.2,"publicationDate":"2016-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68205494","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 will give necessary and sufficient conditions for certain HNN extensions of subgroup separable groups with normal associated subgroup to be conjugacy separable. In fact, we will show that these HNN extensions are conjugacy separable if and only if the normalizer of one of its associated subgroup is conjugacy separable.
{"title":"CONJUGACY SEPARABILITY OF CERTAIN HNN EXTENSIONS WITH NORMAL ASSOCIATED SUBGROUPS","authors":"K. B. Wong, D. Robinson, P. C. Wong","doi":"10.22108/IJGT.2016.9021","DOIUrl":"https://doi.org/10.22108/IJGT.2016.9021","url":null,"abstract":"In this paper, we will give necessary and sufficient conditions for certain HNN extensions of subgroup separable groups with normal associated subgroup to be conjugacy separable. In fact, we will show that these HNN extensions are conjugacy separable if and only if the normalizer of one of its associated subgroup is conjugacy separable.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"5 1","pages":"1-16"},"PeriodicalIF":0.2,"publicationDate":"2016-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68205375","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 solution of the problem of the existence of nontrivial pairs of zero-divisors in group rings of free Burnside groups of sufficiently large odd periods
充分大奇周期自由Burnside群环上非平凡零因子对存在问题的一个正解
{"title":"A remark on group rings of periodic groups","authors":"A. Grigoryan","doi":"10.22108/IJGT.2016.9425","DOIUrl":"https://doi.org/10.22108/IJGT.2016.9425","url":null,"abstract":"A positive solution of the problem of the existence of nontrivial pairs of zero-divisors in group rings of free Burnside groups of sufficiently large odd periods","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"5 1","pages":"23-25"},"PeriodicalIF":0.2,"publicationDate":"2016-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68205504","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 study some relations between the power and quotient power graph of a nite group. These interesting relations motivate us to nd some graph theoretical properties of the quotient power graph and the proper quotient power graph of a nite group G. In addition, we classify those groups whose quotient (proper quotient) power graphs are isomorphic to trees or paths.
{"title":"On groups with specified quotient power graphs","authors":"Mostafa Shaker, M. Iranmanesh","doi":"10.22108/IJGT.2016.8542","DOIUrl":"https://doi.org/10.22108/IJGT.2016.8542","url":null,"abstract":"In this paper we study some relations between the power and quotient power graph of a nite group. These interesting relations motivate us to nd some graph theoretical properties of the quotient power graph and the proper quotient power graph of a nite group G. In addition, we classify those groups whose quotient (proper quotient) power graphs are isomorphic to trees or paths.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"5 1","pages":"49-60"},"PeriodicalIF":0.2,"publicationDate":"2016-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68205652","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}
Darafsheh and Assari in [Normal edge-transitive Cayley graphs on non-abelian groups of order 4p, where p is a prime number, Sci. China Math. 56 (1) (2013) 213 219.] classied the connected normal edge transitive and 1 arc-transitive Cayley graph of groups of order 4p. In this paper we continue this work by classifying the connected Cayley graph of groups of order 2pq, p > q are primes. As a consequence it is proved that Cay(G;S) is a 1 edgetransitive Cayley graph of order 2pq, p > q if and only if jSj is an even integer greater than 2, S = T[ T 1 and T f cba i j 0 i p 1g such that T and T 1 are orbits of Aut(G;S) and
在p为素数的4p阶非abel群上的正规边传递Cayley图中的Darafsheh和Assari。中国数学，56(1)(2013)213 219.使用本文]对4p阶群的连通法向边传递和1弧传递Cayley图进行了分类。在本文中，我们继续这一工作，将2pq, p > q阶群的连通Cayley图分类为素数。因此，证明了Cay(G;S)是2pq阶的1边传递Cayley图，p b> q当且仅当jSj是大于2的偶数，S = T[t1]和tfcba [j] i p 1g，使得T和t1是Aut(G;S)和的轨道
{"title":"Normal edge-transitive and $frac{1}{2}-$arc$-$transitive Cayley graphs on non-abelian groups of order $2pq$, $p > q$ are odd primes","authors":"A. Ashrafi, B. Soleimani","doi":"10.22108/IJGT.2016.6537","DOIUrl":"https://doi.org/10.22108/IJGT.2016.6537","url":null,"abstract":"Darafsheh and Assari in [Normal edge-transitive Cayley graphs on non-abelian groups of order 4p, where p is a prime number, Sci. China Math. 56 (1) (2013) 213 219.] classied the connected normal edge transitive and 1 arc-transitive Cayley graph of groups of order 4p. In this paper we continue this work by classifying the connected Cayley graph of groups of order 2pq, p > q are primes. As a consequence it is proved that Cay(G;S) is a 1 edgetransitive Cayley graph of order 2pq, p > q if and only if jSj is an even integer greater than 2, S = T[ T 1 and T f cba i j 0 i p 1g such that T and T 1 are orbits of Aut(G;S) and","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"5 1","pages":"1-8"},"PeriodicalIF":0.2,"publicationDate":"2016-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68204875","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 group $G$ is said to be a $(PF)C$-group or to have polycyclic-by-finite conjugacy classes, if $G/C_{G}(x^{G})$ is a polycyclic-by-finite group for all $xin G$. This is a generalization of the familiar property of being an $FC$-group. De Falco et al. (respectively, de Giovanni and Trombetti) studied groups whose proper subgroups of infinite rank have finite (respectively, polycyclic) conjugacy classes. Here we consider groups whose proper subgroups of infinite rank are $(PF)C$-groups and we prove that if $G$ is a group of infinite rank having a non-trivial finite or abelian factor group and if all proper subgroups of $G$ of infinite rank are $(PF)C$-groups, then so is $G$. We prove also that if $G$ is a locally soluble-by-finite group of infinite rank which has no simple homomorphic images of infinite rank and whose proper subgroups of infinite rank are $(PF)C$-groups, then so are all proper subgroups of $G$.
{"title":"Groups whose proper subgroups of infinite rank have polycyclic-by-finite conjugacy classes","authors":"Mounia Bouchelaghem, N. Trabelsi","doi":"10.22108/IJGT.2016.8776","DOIUrl":"https://doi.org/10.22108/IJGT.2016.8776","url":null,"abstract":"A group $G$ is said to be a $(PF)C$-group or to have polycyclic-by-finite conjugacy classes, if $G/C_{G}(x^{G})$ is a polycyclic-by-finite group for all $xin G$. This is a generalization of the familiar property of being an $FC$-group. De Falco et al. (respectively, de Giovanni and Trombetti) studied groups whose proper subgroups of infinite rank have finite (respectively, polycyclic) conjugacy classes. Here we consider groups whose proper subgroups of infinite rank are $(PF)C$-groups and we prove that if $G$ is a group of infinite rank having a non-trivial finite or abelian factor group and if all proper subgroups of $G$ of infinite rank are $(PF)C$-groups, then so is $G$. We prove also that if $G$ is a locally soluble-by-finite group of infinite rank which has no simple homomorphic images of infinite rank and whose proper subgroups of infinite rank are $(PF)C$-groups, then so are all proper subgroups of $G$.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"5 1","pages":"61-67"},"PeriodicalIF":0.2,"publicationDate":"2016-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68205249","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}
For a symmetric group G:=symn">G:=symnG:=symn and a conjugacy class X">XX of involutions in G">GG, it is known that if the class of involutions does not have a unique fixed point, then - with a few small exceptions - given two elements a,x∈X">a,x∈Xa,x∈X, either ⟨a,x⟩">⟨a,x⟩⟨a,x⟩ is isomorphic to the dihedral group D8">D8D8, or there is a further element y∈X">y∈Xy∈X such that ⟨a,y⟩≅⟨x,y⟩≅D8">⟨a,y⟩≅⟨x,y⟩≅D8⟨a,y⟩≅⟨x,y⟩≅D8 (P. Rowley and D. Ward, On π">ππ-Product Involution Graphs in Symmetric Groups. MIMS ePrint, 2014). One natural generalisation of this to p">pp-elements is to consider when two conjugate p">pp-elements generate a wreath product of two cyclic groups of order p">pp. In this paper we give necessary and sufficient conditions for this in the case that our p">pp-elements have full support. These conditions relate to given matrices that are of circulant or permutation type, and corresponding polynomials that represent these matrices. We also consider the case that the elements do not have full support, and see why generalising our results to such elements would not be a natural generalisation.
{"title":"Conjugate $p$-elements of full support that generate the wreath product $C_{p}wr C_{p}$","authors":"David Ward","doi":"10.22108/IJGT.2016.7806","DOIUrl":"https://doi.org/10.22108/IJGT.2016.7806","url":null,"abstract":"For a symmetric group G:=symn\">G:=symnG:=symn and a conjugacy class X\">XX of involutions in G\">GG, it is known that if the class of involutions does not have a unique fixed point, then - with a few small exceptions - given two elements a,x∈X\">a,x∈Xa,x∈X, either ⟨a,x⟩\">⟨a,x⟩⟨a,x⟩ is isomorphic to the dihedral group D8\">D8D8, or there is a further element y∈X\">y∈Xy∈X such that ⟨a,y⟩≅⟨x,y⟩≅D8\">⟨a,y⟩≅⟨x,y⟩≅D8⟨a,y⟩≅⟨x,y⟩≅D8 (P. Rowley and D. Ward, On π\">ππ-Product Involution Graphs in Symmetric Groups. MIMS ePrint, 2014). One natural generalisation of this to p\">pp-elements is to consider when two conjugate p\">pp-elements generate a wreath product of two cyclic groups of order p\">pp. In this paper we give necessary and sufficient conditions for this in the case that our p\">pp-elements have full support. These conditions relate to given matrices that are of circulant or permutation type, and corresponding polynomials that represent these matrices. We also consider the case that the elements do not have full support, and see why generalising our results to such elements would not be a natural generalisation.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"5 1","pages":"9-35"},"PeriodicalIF":0.2,"publicationDate":"2016-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68205180","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 commutativity degree, Pr(G)">Pr(G)Pr(G), of a finite group G">GG (i.e. the probability that two (randomly chosen) elements of G">GGcommute with respect to its operation)) has been studied well by many authors. It is well-known that the best upper bound for Pr(G)">Pr(G)Pr(G) is 58">5858 for a finite non-abelian group G">GG. In this paper, we will define the same concept for a finite non--abelian Moufang loop M">MM and try to give a best upper bound for Pr(M)">Pr(M)Pr(M). We will prove that for a well-known class of finite Moufang loops, named Chein loops, and its modifications, this best upper bound is 2332">23322332. So, our conjecture is that for any finite Moufang loop M">MM, Pr(M)≤2332">Pr(M)≤2332Pr(M)≤2332. Also, we will obtain some results related to the Pr(M)">Pr(M)Pr(M) and ask the similar questions raised and answered in group theory about the relations between the structure of a finite group and its commutativity degree in finite Moufang loops.
{"title":"ON THE COMMUTATIVITY DEGREE IN FINITE MOUFANG LOOPS","authors":"K. Ahmadidelir","doi":"10.22108/IJGT.2016.8477","DOIUrl":"https://doi.org/10.22108/IJGT.2016.8477","url":null,"abstract":"The commutativity degree, Pr(G)\">Pr(G)Pr(G), of a finite group G\">GG (i.e. the probability that two (randomly chosen) elements of G\">GGcommute with respect to its operation)) has been studied well by many authors. It is well-known that the best upper bound for Pr(G)\">Pr(G)Pr(G) is 58\">5858 for a finite non-abelian group G\">GG. In this paper, we will define the same concept for a finite non--abelian Moufang loop M\">MM and try to give a best upper bound for Pr(M)\">Pr(M)Pr(M). We will prove that for a well-known class of finite Moufang loops, named Chein loops, and its modifications, this best upper bound is 2332\">23322332. So, our conjecture is that for any finite Moufang loop M\">MM, Pr(M)≤2332\">Pr(M)≤2332Pr(M)≤2332. Also, we will obtain some results related to the Pr(M)\">Pr(M)Pr(M) and ask the similar questions raised and answered in group theory about the relations between the structure of a finite group and its commutativity degree in finite Moufang loops.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"5 1","pages":"37-47"},"PeriodicalIF":0.2,"publicationDate":"2016-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68204979","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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