Pub Date : 2020-01-28DOI: 10.22108/IJGT.2020.120894.1593
Ö. Deveci, E. Karaduman
Deveci et al. defined 6. the Fibonacci-circulant sequences of the first and second kinds as shown, respectively: x_{n}¹=-x_{n-1}¹+x_{n-2}¹-x_{n-3}¹ for n≥4, where x₁¹=x₂¹=0 and x₃¹=1and x_{n}²=-x_{n-3}²-x_{n-4}²+x_{n-5}² for n≥6, where x₁²=x₂²=x₃²=x₄²=0 and x₅²=1.Also, they extended the Fibonacci-circulant sequences of the first and second kinds to groups. In this work, we obtain the periods of the Fibonacci-circulant sequences of the first and second kinds in the binary polyhedral groups.
{"title":"The Fibonacci-Circulant Sequences in the Binary Polyhedral Groups","authors":"Ö. Deveci, E. Karaduman","doi":"10.22108/IJGT.2020.120894.1593","DOIUrl":"https://doi.org/10.22108/IJGT.2020.120894.1593","url":null,"abstract":"Deveci et al. defined 6. the Fibonacci-circulant sequences of the first and second kinds as shown, respectively: x_{n}¹=-x_{n-1}¹+x_{n-2}¹-x_{n-3}¹ for n≥4, where x₁¹=x₂¹=0 and x₃¹=1and x_{n}²=-x_{n-3}²-x_{n-4}²+x_{n-5}² for n≥6, where x₁²=x₂²=x₃²=x₄²=0 and x₅²=1.Also, they extended the Fibonacci-circulant sequences of the first and second kinds to groups. In this work, we obtain the periods of the Fibonacci-circulant sequences of the first and second kinds in the binary polyhedral groups.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45830552","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 : 2019-12-16DOI: 10.22108/IJGT.2019.119105.1570
I. Malinowska
P. Hall proved that a finite group G is supersoluble with elementary abelian Sylow sub-groups if and only if every subgroup of G is complemented in G. He called such groups complemented. A. Ballester-Bolinches and X. Guo established the structure of minimal non-complemented groups. We give the classification of finite non-soluble groups all of whose second maximal subgroups are complemented groups. We also prove that every finite group with less than 21 non-complemented non-minimal {2; 3; 5}--subgroups is soluble.
{"title":"Influence of complemented subgroups on the structure of finite groups","authors":"I. Malinowska","doi":"10.22108/IJGT.2019.119105.1570","DOIUrl":"https://doi.org/10.22108/IJGT.2019.119105.1570","url":null,"abstract":"P. Hall proved that a finite group G is supersoluble with elementary abelian Sylow sub-groups if and only if every subgroup of G is complemented in G. He called such groups complemented. A. Ballester-Bolinches and X. Guo established the structure of minimal non-complemented groups. We give the classification of finite non-soluble groups all of whose second maximal subgroups are complemented groups. We also prove that every finite group with less than 21 non-complemented non-minimal {2; 3; 5}--subgroups is soluble.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2019-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48426271","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 : 2019-12-01DOI: 10.22108/IJGT.2018.110074.1472
Julian Brough
We say that a group $G$ satisfies the one-prime power hypothesis for conjugacy classes if the greatest common divisor for all pairs of distinct conjugacy class sizes are prime powers. Insoluble groups which satisfy the one-prime power hypothesis have been classified. However it has remained an open question whether the one-prime power hypothesis is inherited by normal subgroups and quotients groups. In this note we construct examples to show the one-prime power hypothesis is not necessarily inherited by normal subgroups or quotient groups.
{"title":"The one-prime power hypothesis for conjugacy classes restricted to normal subgroups and quotient groups","authors":"Julian Brough","doi":"10.22108/IJGT.2018.110074.1472","DOIUrl":"https://doi.org/10.22108/IJGT.2018.110074.1472","url":null,"abstract":"We say that a group $G$ satisfies the one-prime power hypothesis for conjugacy classes if the greatest common divisor for all pairs of distinct conjugacy class sizes are prime powers. Insoluble groups which satisfy the one-prime power hypothesis have been classified. However it has remained an open question whether the one-prime power hypothesis is inherited by normal subgroups and quotients groups. In this note we construct examples to show the one-prime power hypothesis is not necessarily inherited by normal subgroups or quotient groups.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"8 1","pages":"1-3"},"PeriodicalIF":0.2,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41950092","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 : 2019-09-01DOI: 10.22108/IJGT.2018.108302.1458
S. M. Robati
Let $G$ be a finite group. In this paper, we study the structure of finite groups having $|G|-r$ cyclic subgroups for $3leq rleq 5$.
设$G$是一个有限群。本文研究了$3leq rleq 5$具有$|G|-r$循环子群的有限群结构。
{"title":"On finite groups having a certain number of cyclic subgroups","authors":"S. M. Robati","doi":"10.22108/IJGT.2018.108302.1458","DOIUrl":"https://doi.org/10.22108/IJGT.2018.108302.1458","url":null,"abstract":"Let $G$ be a finite group. In this paper, we study the structure of finite groups having $|G|-r$ cyclic subgroups for $3leq rleq 5$.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"8 1","pages":"1-8"},"PeriodicalIF":0.2,"publicationDate":"2019-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42748612","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 : 2019-09-01DOI: 10.22108/IJGT.2018.111238.1478
Alireza Rahimipour, Yousof Farzaneh
A finite group $G$ has uniform spread $k$ if there exists a fixed conjugacy class $C$ of elements in $G$ with the property that for any $k$ nontrivial elements $s_1, s_2,ldots,s_k$ in $G$ there exists $yin C$ such that $G = langle s_i,yrangle$ for $i=1, 2,ldots,k$. Further, the exact uniform spread of $G$ is the largest $k$ such that $G$ has the uniform spread $k$. In this paper we give upper bounds on the exact uniform spreads of thirteen sporadic simple groups.
{"title":"Upper bounds on the uniform spreads of the sporadic simple groups","authors":"Alireza Rahimipour, Yousof Farzaneh","doi":"10.22108/IJGT.2018.111238.1478","DOIUrl":"https://doi.org/10.22108/IJGT.2018.111238.1478","url":null,"abstract":"A finite group $G$ has uniform spread $k$ if there exists a fixed conjugacy class $C$ of elements in $G$ with the property that for any $k$ nontrivial elements $s_1, s_2,ldots,s_k$ in $G$ there exists $yin C$ such that $G = langle s_i,yrangle$ for $i=1, 2,ldots,k$. Further, the exact uniform spread of $G$ is the largest $k$ such that $G$ has the uniform spread $k$. In this paper we give upper bounds on the exact uniform spreads of thirteen sporadic simple groups.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"8 1","pages":"15-31"},"PeriodicalIF":0.2,"publicationDate":"2019-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44228879","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 : 2019-09-01DOI: 10.22108/IJGT.2018.104745.1437
Abdulsatar J. AL-Juburie, A. Duncan
Let GΓ be a partially commutative group. We find a finite presentation for the subgroupConjV(GΓ) of compressed vertex conjugating automorphisms of the automorphism group Aut(GΓ) of G. We have written GAP packages which compute presentations for Aut(GΓ) and its subgroups Conj(GΓ) and ConjV(GΓ).
{"title":"A presentation for the subgroup of compressed conjugating automorphisms of a partially commutative group","authors":"Abdulsatar J. AL-Juburie, A. Duncan","doi":"10.22108/IJGT.2018.104745.1437","DOIUrl":"https://doi.org/10.22108/IJGT.2018.104745.1437","url":null,"abstract":"Let GΓ be a partially commutative group. We find a finite presentation for the subgroupConjV(GΓ) of compressed vertex conjugating automorphisms of the automorphism group Aut(GΓ) of G. We have written GAP packages which compute presentations for Aut(GΓ) and its subgroups Conj(GΓ) and ConjV(GΓ).","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"8 1","pages":"33-42"},"PeriodicalIF":0.2,"publicationDate":"2019-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45732318","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 : 2019-08-14DOI: 10.22108/IJGT.2019.115910.1538
Yadollah Marefat, Maghsoud Jahani, H. Refaghat, Bahram Vakili Fasaghandisi
Let G be a finite group and ψ(G) = ∑ g∈G o(g), where o(g) denotes the order of g ∈ G. We show that the Conjecture 4.6.5 posed in [Group Theory and Computation, (2018) 59-90], is incorrect. In fact, we find a pair of finite groups G and S of the same order such that ψ(G) < ψ(S), with G solvable and S simple.
{"title":"The minimum sum of element orders of finite groups","authors":"Yadollah Marefat, Maghsoud Jahani, H. Refaghat, Bahram Vakili Fasaghandisi","doi":"10.22108/IJGT.2019.115910.1538","DOIUrl":"https://doi.org/10.22108/IJGT.2019.115910.1538","url":null,"abstract":"Let G be a finite group and ψ(G) = ∑ g∈G o(g), where o(g) denotes the order of g ∈ G. We show that the Conjecture 4.6.5 posed in [Group Theory and Computation, (2018) 59-90], is incorrect. In fact, we find a pair of finite groups G and S of the same order such that ψ(G) < ψ(S), with G solvable and S simple.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2019-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47226543","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 : 2019-08-14DOI: 10.22108/IJGT.2019.116209.1543
B. Taeri, Fatemeh Tayanloo-Beyg
We characterize finite non-nilpotent groups $G$ with a unique non-nilpotent proper subgroup. We show that $|G|$ has at most three prime divisors. When $G$ is supersolvable we find the presentation of $G$ and when $G$ is non-supersolvable we show that either $G$ is a direct product of an Schmidt group and a cyclic group or a semi direct product of a $p$-group by a cyclic group of prime power order.
{"title":"Characterization of finite groups with a unique non-nilpotent proper subgroup","authors":"B. Taeri, Fatemeh Tayanloo-Beyg","doi":"10.22108/IJGT.2019.116209.1543","DOIUrl":"https://doi.org/10.22108/IJGT.2019.116209.1543","url":null,"abstract":"We characterize finite non-nilpotent groups $G$ with a unique non-nilpotent proper subgroup. We show that $|G|$ has at most three prime divisors. When $G$ is supersolvable we find the presentation of $G$ and when $G$ is non-supersolvable we show that either $G$ is a direct product of an Schmidt group and a cyclic group or a semi direct product of a $p$-group by a cyclic group of prime power order.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2019-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45216386","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 : 2019-08-13DOI: 10.22108/IJGT.2019.116275.1545
W. Cocke
In this note we prove an analog of Hilbert’s theorem 90 for finite nilpotent groups. Our version of Hilbert’s theorem 90 was inspired by the Boston–Bush–Hajir (BBH) heuristics in number theory and will be useful in extending the BBH heuristics beyond quadratic field extensions.
{"title":"Hilbert's Theorem 90 for Finite Nilpotent Groups","authors":"W. Cocke","doi":"10.22108/IJGT.2019.116275.1545","DOIUrl":"https://doi.org/10.22108/IJGT.2019.116275.1545","url":null,"abstract":"In this note we prove an analog of Hilbert’s theorem 90 for finite nilpotent groups. Our version of Hilbert’s theorem 90 was inspired by the Boston–Bush–Hajir (BBH) heuristics in number theory and will be useful in extending the BBH heuristics beyond quadratic field extensions.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2019-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48029841","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 : 2019-07-25DOI: 10.22108/IJGT.2019.116399.1546
T. H. Dung
Let D be a division ring with center F and assume that N is a locally soluble almost subnormal subgroup of the multiplicative group D∗ of D. We prove that if N is algebraic over F , then N is central. This answers partially [11, Conjecture 1].
{"title":"A note on locally soluble almost subnormal subgroups in divsion rings","authors":"T. H. Dung","doi":"10.22108/IJGT.2019.116399.1546","DOIUrl":"https://doi.org/10.22108/IJGT.2019.116399.1546","url":null,"abstract":"Let D be a division ring with center F and assume that N is a locally soluble almost subnormal subgroup of the multiplicative group D∗ of D. We prove that if N is algebraic over F , then N is central. This answers partially [11, Conjecture 1].","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2019-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44793290","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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