Pub Date : 2017-10-21DOI: 10.22108/IJGT.2017.21975
A. Shevlyakov
We study equations over completely simple semigroups and describe the coordinate semigroups of irreducible algebraic sets for such semigroups.
我们研究了完全简单半群上的方程，并描述了这类半群的不可约代数集的坐标半群。
{"title":"On algebraic geometry over completely simple semigroups","authors":"A. Shevlyakov","doi":"10.22108/IJGT.2017.21975","DOIUrl":"https://doi.org/10.22108/IJGT.2017.21975","url":null,"abstract":"We study equations over completely simple semigroups and describe the coordinate semigroups of irreducible algebraic sets for such semigroups.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"8 1","pages":"1-10"},"PeriodicalIF":0.2,"publicationDate":"2017-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49615447","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 : 2017-09-01DOI: 10.22108/IJGT.2017.12043
A. Camina, R. Camina
A finite group $G$ satisfies the on-prime power hypothesis for conjugacy class sizes if any two conjugacy class sizes $m$ and $n$ are either equal or have a common divisor a prime power. Taeri conjectured that an insoluble group satisfying this condition is isomorphic to $S times A$ where $A$ is abelian and $S cong PSL_2(q)$ for $q in {4,8}$. We confirm this conjecture.
{"title":"One-prime power hypothesis for conjugacy class sizes","authors":"A. Camina, R. Camina","doi":"10.22108/IJGT.2017.12043","DOIUrl":"https://doi.org/10.22108/IJGT.2017.12043","url":null,"abstract":"A finite group $G$ satisfies the on-prime power hypothesis for conjugacy class sizes if any two conjugacy class sizes $m$ and $n$ are either equal or have a common divisor a prime power. Taeri conjectured that an insoluble group satisfying this condition is isomorphic to $S times A$ where $A$ is abelian and $S cong PSL_2(q)$ for $q in {4,8}$. We confirm this conjecture.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"6 1","pages":"13-19"},"PeriodicalIF":0.2,"publicationDate":"2017-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42688540","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 : 2017-09-01DOI: 10.22108/IJGT.2017.21213
H. Abdolzadeh, Reza Sabzchi
We determine a new infinite sequence of finite $2$-groups with deficiency zero. The groups have $2$ generators and $2$ relations, they have coclass $3$ and they are not metacyclic.
{"title":"An infinite family of finite $2$-groups with deficiency zero","authors":"H. Abdolzadeh, Reza Sabzchi","doi":"10.22108/IJGT.2017.21213","DOIUrl":"https://doi.org/10.22108/IJGT.2017.21213","url":null,"abstract":"We determine a new infinite sequence of finite $2$-groups with deficiency zero. The groups have $2$ generators and $2$ relations, they have coclass $3$ and they are not metacyclic.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"39 6","pages":"45-49"},"PeriodicalIF":0.2,"publicationDate":"2017-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41285545","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 : 2017-08-17DOI: 10.22108/IJGT.2017.21613
D. Crnković, Andrea Švob
In this paper we construct transitive $t$-designs from the linear groups $L(2,q), q leq 23$. Thereby we classify $t$-designs, $t ge 2$, admitting a transitive action of the linear groups $L(2,q), q leq 23$, up to 35 points and obtained numerous transitive designs, for $36leq vleq 55$. In many cases we proved the existence of $t$-designs with certain parameter sets. Among others we constructed $t$-designs with parameters $2$-$(55,10,4)$, $3$-$(24,11,495)$, $3$-$(24,12, 5m), m in {11, 12,22, 33, 44, 66, 132}$. Furthermore, we constructed strongly regular graphs admitting a transitive action of the linear groups $L(2,q), q leq 23$.
{"title":"Transitive $t$-designs and strongly regular graphs constructed from linear groups $L(2,q)$, $q leq 23$","authors":"D. Crnković, Andrea Švob","doi":"10.22108/IJGT.2017.21613","DOIUrl":"https://doi.org/10.22108/IJGT.2017.21613","url":null,"abstract":"In this paper we construct transitive $t$-designs from the linear groups $L(2,q), q leq 23$. Thereby we classify $t$-designs, $t ge 2$, admitting a transitive action of the linear groups $L(2,q), q leq 23$, up to 35 points and obtained numerous transitive designs, for $36leq vleq 55$. In many cases we proved the existence of $t$-designs with certain parameter sets. Among others we constructed $t$-designs with parameters $2$-$(55,10,4)$, $3$-$(24,11,495)$, $3$-$(24,12, 5m), m in {11, 12,22, 33, 44, 66, 132}$. Furthermore, we constructed strongly regular graphs admitting a transitive action of the linear groups $L(2,q), q leq 23$.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"8 1","pages":"43-64"},"PeriodicalIF":0.2,"publicationDate":"2017-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47440469","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 : 2017-06-09DOI: 10.22108/IJGT.2017.21482
D. Levy
{"title":"Sylow multiplicities in finite groups","authors":"D. Levy","doi":"10.22108/IJGT.2017.21482","DOIUrl":"https://doi.org/10.22108/IJGT.2017.21482","url":null,"abstract":"","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"7 1","pages":"1-8"},"PeriodicalIF":0.2,"publicationDate":"2017-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41380259","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 : 2017-06-09DOI: 10.22108/IJGT.2017.21481
Nil Mansuroğlu
Let $L$ be a free Lie algebra of rank $rgeq2$ over a field $F$ and let $L_n$ denote the degree $n$ homogeneous component of $L$. By using the dimensions of the corresponding homogeneous and fine homogeneous components of the second derived ideal of free centre-by-metabelian Lie algebra over a field $F$, we determine the dimension of $[L_2,L_2,L_1]$. Moreover, by this method, we show that the dimension of $[L_2,L_2,L_1]$ over a field of characteristic $2$ is different from the dimension over a field of characteristic other than $2$.
{"title":"On the dimension of the product $[L_2,L_2,L_1]$ in free Lie algebras","authors":"Nil Mansuroğlu","doi":"10.22108/IJGT.2017.21481","DOIUrl":"https://doi.org/10.22108/IJGT.2017.21481","url":null,"abstract":"Let $L$ be a free Lie algebra of rank $rgeq2$ over a field $F$ and let $L_n$ denote the degree $n$ homogeneous component of $L$. By using the dimensions of the corresponding homogeneous and fine homogeneous components of the second derived ideal of free centre-by-metabelian Lie algebra over a field $F$, we determine the dimension of $[L_2,L_2,L_1]$. Moreover, by this method, we show that the dimension of $[L_2,L_2,L_1]$ over a field of characteristic $2$ is different from the dimension over a field of characteristic other than $2$.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"7 1","pages":"45-50"},"PeriodicalIF":0.2,"publicationDate":"2017-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46336082","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 : 2017-06-01DOI: 10.22108/IJGT.2017.12406
Mahin Ranjbari, Y. Zamani
In this paper, we give a necessary and sufficient condition for the equality of two symmetrized decomposable polynomials. Then, we study some algebraic and geometric properties of the induced operators over symmetry classes of polynomials in the case of linear characters.
{"title":"Induced operators on symmetry classes of polynomials","authors":"Mahin Ranjbari, Y. Zamani","doi":"10.22108/IJGT.2017.12406","DOIUrl":"https://doi.org/10.22108/IJGT.2017.12406","url":null,"abstract":"In this paper, we give a necessary and sufficient condition for the equality of two symmetrized decomposable polynomials. Then, we study some algebraic and geometric properties of the induced operators over symmetry classes of polynomials in the case of linear characters.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"6 1","pages":"21-35"},"PeriodicalIF":0.2,"publicationDate":"2017-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45438156","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 : 2017-06-01DOI: 10.22108/IJGT.2017.13314
J. B. Nganou
A positive integer $n$ is called a CLT number if every group of order $n$ satisfies the converse of Lagrange's Theorem. In this note, we find all CLT and supersolvable numbers up to $1000$. We also formulate some questions about the distribution of these numbers.
{"title":"Converse of Lagrange's theorem (CLT) numbers under $1000$","authors":"J. B. Nganou","doi":"10.22108/IJGT.2017.13314","DOIUrl":"https://doi.org/10.22108/IJGT.2017.13314","url":null,"abstract":"A positive integer $n$ is called a CLT number if every group of order $n$ satisfies the converse of Lagrange's Theorem. In this note, we find all CLT and supersolvable numbers up to $1000$. We also formulate some questions about the distribution of these numbers.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"6 1","pages":"37-42"},"PeriodicalIF":0.2,"publicationDate":"2017-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47100518","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 : 2017-06-01DOI: 10.22108/IJGT.2017.11176
M. Zarrin
Let $G$ be a group and $mathcal{N}$ be the class of all nilpotent groups. A subset $A$ of $G$ is said to be nonnilpotent if for any two distinct elements $a$ and $b$ in $A$, $langle a, brangle notin mathcal{N}$. If, for any other nonnilpotent subset $B$ in $G$, $|A|geq |B|$, then $A$ is said to be a maximal nonnilpotent subset and the cardinality of this subset (if it exists) is denoted by $omega(mathcal{N}_G)$. In this paper, among other results, we obtain $omega(mathcal{N}_{Suz(q)})$ and $omega(mathcal{N}_{PGL(2,q)})$, where $Suz(q)$ is the Suzuki simple group over the field with $q$ elements and $PGL(2,q)$ is the projective general linear group of degree $2$ over the finite field with $q$ elements, respectively.
{"title":"Nonnilpotent subsets in the Suzuki groups","authors":"M. Zarrin","doi":"10.22108/IJGT.2017.11176","DOIUrl":"https://doi.org/10.22108/IJGT.2017.11176","url":null,"abstract":"Let $G$ be a group and $mathcal{N}$ be the class of all nilpotent groups. A subset $A$ of $G$ is said to be nonnilpotent if for any two distinct elements $a$ and $b$ in $A$, $langle a, brangle notin mathcal{N}$. If, for any other nonnilpotent subset $B$ in $G$, $|A|geq |B|$, then $A$ is said to be a maximal nonnilpotent subset and the cardinality of this subset (if it exists) is denoted by $omega(mathcal{N}_G)$. In this paper, among other results, we obtain $omega(mathcal{N}_{Suz(q)})$ and $omega(mathcal{N}_{PGL(2,q)})$, where $Suz(q)$ is the Suzuki simple group over the field with $q$ elements and $PGL(2,q)$ is the projective general linear group of degree $2$ over the finite field with $q$ elements, respectively.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"6 1","pages":"7-15"},"PeriodicalIF":0.2,"publicationDate":"2017-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46382699","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 : 2017-06-01DOI: 10.22108/IJGT.2017.12396
Cui Zhang
The Theorem 12 in [A note on $p$-nilpotence and solvability of finite groups, J. Algebra 321 (2009) 1555--1560.] investigated the non-abelian simple groups in which some maximal subgroups have primes indices. In this note we show that this result can be applied to prove that the finite groups in which every non-nilpotent maximal subgroup has prime index are solvable.
{"title":"A note on finite groups with the indice of some maximal subgroups being primes","authors":"Cui Zhang","doi":"10.22108/IJGT.2017.12396","DOIUrl":"https://doi.org/10.22108/IJGT.2017.12396","url":null,"abstract":"The Theorem 12 in [A note on $p$-nilpotence and solvability of finite groups, J. Algebra 321 (2009) 1555--1560.] investigated the non-abelian simple groups in which some maximal subgroups have primes indices. In this note we show that this result can be applied to prove that the finite groups in which every non-nilpotent maximal subgroup has prime index are solvable.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"6 1","pages":"17-20"},"PeriodicalIF":0.2,"publicationDate":"2017-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45546099","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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