Pub Date : 2022-03-01DOI: 10.22108/IJGT.2021.120906.1594
M. Jahandideh, M. Darafsheh
Let $G$ be a group and $omega(G)={o(g)|gin G}$ be the set of element orders of $G$. Let $kinomega(G)$ and $s_{k}=|{gin G|o(g)=k}|$. Let $nse(G)={s_{k}|kinomega(G)}.$ In this paper, we prove that if $G$ is a group and $G_{2}(5)$ is the Chevalley simple group of type $G_{2}$ over $GF(5)$ such that $nse(G)=nse(G_{2}(5))$, then $Gcong G_{2}(5)$.
{"title":"Characterization of the Chevalley group $G_{2}(5)$ by the set of numbers of the same order elements","authors":"M. Jahandideh, M. Darafsheh","doi":"10.22108/IJGT.2021.120906.1594","DOIUrl":"https://doi.org/10.22108/IJGT.2021.120906.1594","url":null,"abstract":"Let $G$ be a group and $omega(G)={o(g)|gin G}$ be the set of element orders of $G$. Let $kinomega(G)$ and $s_{k}=|{gin G|o(g)=k}|$. Let $nse(G)={s_{k}|kinomega(G)}.$ In this paper, we prove that if $G$ is a group and $G_{2}(5)$ is the Chevalley simple group of type $G_{2}$ over $GF(5)$ such that $nse(G)=nse(G_{2}(5))$, then $Gcong G_{2}(5)$.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"11 1","pages":"7-16"},"PeriodicalIF":0.2,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43066884","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 : 2021-12-01DOI: 10.22108/IJGT.2020.123287.1625
Zahara Bahrami, B. Taeri
Let $G$ be a finite group which is not cyclic of prime power order. The join graph $Delta(G)$ of $G$ is a graph whose vertex set is the set of all proper subgroups of $G$, which are not contained in the Frattini subgroup $G$ and two distinct vertices $H$ and $K$ are adjacent if and only if $G=langle H, Krangle$. Among other results, we show that if $G$ is a finite cyclic group and $H$ is a finite group such that $Delta(G)congDelta(H)$, then $H$ is cyclic. Also we prove that $Delta(G)congDelta(A_5)$ if and only if $Gcong A_5$.
{"title":"Some results on the join graph of finite groups","authors":"Zahara Bahrami, B. Taeri","doi":"10.22108/IJGT.2020.123287.1625","DOIUrl":"https://doi.org/10.22108/IJGT.2020.123287.1625","url":null,"abstract":"Let $G$ be a finite group which is not cyclic of prime power order. The join graph $Delta(G)$ of $G$ is a graph whose vertex set is the set of all proper subgroups of $G$, which are not contained in the Frattini subgroup $G$ and two distinct vertices $H$ and $K$ are adjacent if and only if $G=langle H, Krangle$. Among other results, we show that if $G$ is a finite cyclic group and $H$ is a finite group such that $Delta(G)congDelta(H)$, then $H$ is cyclic. Also we prove that $Delta(G)congDelta(A_5)$ if and only if $Gcong A_5$.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"10 1","pages":"175-186"},"PeriodicalIF":0.2,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46457834","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 : 2021-10-25DOI: 10.22108/IJGT.2021.126694.1664
M. Salih, M. Haval
Let R be a non trivial finite commutative ring with identity and G be a non trivial group. We denote by P(RG) the probability that the product of two randomly chosen elements of a finite group ring RG is zero. We show that P(RG) <0.25 if and only if RG is not isomorphic to Z2C2, Z3C2, Z2C3. Furthermore, we give the upper bound and lower bound for P(RG). In particular, we present the general formula for P(RG), where R is a finite field of characteristic p and |G| ≤ 4.
{"title":"On the probability of zero divisor elements in group rings","authors":"M. Salih, M. Haval","doi":"10.22108/IJGT.2021.126694.1664","DOIUrl":"https://doi.org/10.22108/IJGT.2021.126694.1664","url":null,"abstract":"Let R be a non trivial finite commutative ring with identity and G be a non trivial\u0000group. We denote by P(RG) the probability that the product of two randomly chosen\u0000elements of a finite group ring RG is zero. We show that P(RG) <0.25 if and only if\u0000RG is not isomorphic to Z2C2, Z3C2, Z2C3. Furthermore, we give the upper bound and lower bound for\u0000P(RG). In particular, we present the general formula for P(RG), where R is a finite field of\u0000characteristic p and |G| ≤ 4.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44489952","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 : 2021-09-29DOI: 10.22108/IJGT.2021.129788.1732
M. Saha, Sucharita Biswas, Angsuman Das
The co-maximal subgroup graph $Gamma(G)$ of a group $G$ is a graph whose vertices are non-trivial proper subgroups of $G$ and two vertices $H$ and $K$ are adjacent if $HK = G$. In this paper, we study and characterize various properties like diameter, domination number, perfectness, hamiltonicity, etc. of $Gamma(mathbb{Z}_n)$.
{"title":"On Co-Maximal Subgroup Graph of $Z_n$","authors":"M. Saha, Sucharita Biswas, Angsuman Das","doi":"10.22108/IJGT.2021.129788.1732","DOIUrl":"https://doi.org/10.22108/IJGT.2021.129788.1732","url":null,"abstract":"The co-maximal subgroup graph $Gamma(G)$ of a group $G$ is a graph whose vertices are non-trivial proper subgroups of $G$ and two vertices $H$ and $K$ are adjacent if $HK = G$. In this paper, we study and characterize various properties like diameter, domination number, perfectness, hamiltonicity, etc. of $Gamma(mathbb{Z}_n)$.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42730354","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 : 2021-09-24DOI: 10.22108/IJGT.2021.129515.1700
M. Ferrara, M. Trombetti
We prove that the following families of (infinite) groups have complemented subgroup lattice: alternating groups, finitary symmetric groups, Suzuki groups over an infinite locally finite field of characteristic $2$, Ree groups over an infinite locally finite field of characteristic~$3$. We also show that if the Sylow primary subgroups of a locally finite simple group $G$ have complemented subgroup lattice, then this is also the case for $G$.
{"title":"Infinite Locally Finite Simple Groups with Many Complemented Subgroups","authors":"M. Ferrara, M. Trombetti","doi":"10.22108/IJGT.2021.129515.1700","DOIUrl":"https://doi.org/10.22108/IJGT.2021.129515.1700","url":null,"abstract":"We prove that the following families of (infinite) groups have complemented subgroup lattice: alternating groups, finitary symmetric groups, Suzuki groups over an infinite locally finite field of characteristic $2$, Ree groups over an infinite locally finite field of characteristic~$3$. We also show that if the Sylow primary subgroups of a locally finite simple group $G$ have complemented subgroup lattice, then this is also the case for $G$.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42566725","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 : 2021-09-24DOI: 10.22108/IJGT.2021.128455.1690
Yong Yang
Heineken [H. Heineken, Nilpotent subgroups of finite soluble groups, Arch. Math.(Basel), 56 no. 5 (1991) 417--423.] studied the order of the nilpotent subgroups of the largest order of a solvable group. We point out an error, and thus refute the proof of the main result of [H. Heineken, Nilpotent subgroups of finite soluble groups, Arch. Math.(Basel)}, 56 no. 5 (1991) 417--423.].
{"title":"On a result of nilpotent subgroups of solvable groups","authors":"Yong Yang","doi":"10.22108/IJGT.2021.128455.1690","DOIUrl":"https://doi.org/10.22108/IJGT.2021.128455.1690","url":null,"abstract":"Heineken [H. Heineken, Nilpotent subgroups of finite soluble groups, Arch. Math.(Basel), 56 no. 5 (1991) 417--423.] studied the order of the nilpotent subgroups of the largest order of a solvable group. We point out an error, and thus refute the proof of the main result of [H. Heineken, Nilpotent subgroups of finite soluble groups, Arch. Math.(Basel)}, 56 no. 5 (1991) 417--423.].","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43049801","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 : 2021-09-01DOI: 10.22108/IJGT.2020.112121.1489
Jessica Hamm, A. Way
There are many different graphs one can associate to a group. Some examples are the well-known Cayley graph, the zero divisor graph (of a ring), the power graph, and the recently introduced coprime graph of a group. The coprime graph of a group $G$, denoted $Gamma_G$, is the graph whose vertices are the group elements with $g$ adjacent to $h$ if and only if $(o(g),o(h))=1$. In this paper we calculate the independence number of the coprime graph of the dihedral groups. Additionally, we characterize the groups whose coprime graph is perfect.
{"title":"Parameters of the coprime graph of a group","authors":"Jessica Hamm, A. Way","doi":"10.22108/IJGT.2020.112121.1489","DOIUrl":"https://doi.org/10.22108/IJGT.2020.112121.1489","url":null,"abstract":"There are many different graphs one can associate to a group. Some examples are the well-known Cayley graph, the zero divisor graph (of a ring), the power graph, and the recently introduced coprime graph of a group. The coprime graph of a group $G$, denoted $Gamma_G$, is the graph whose vertices are the group elements with $g$ adjacent to $h$ if and only if $(o(g),o(h))=1$. In this paper we calculate the independence number of the coprime graph of the dihedral groups. Additionally, we characterize the groups whose coprime graph is perfect.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"10 1","pages":"137-147"},"PeriodicalIF":0.2,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46494227","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 : 2021-08-15DOI: 10.22108/IJGT.2021.130057.1735
L. A. Kurdachenko, A. A. Pypka, I. Subbotin
We study the automorphism groups of finite-dimensional cyclic Leibniz algebras. In this connection, we consider the relationships between groups, modules over associative rings and Leibniz algebras.
{"title":"ON THE AUTOMORPHISM GROUPS OF SOME LEIBNIZ ALGEBRAS","authors":"L. A. Kurdachenko, A. A. Pypka, I. Subbotin","doi":"10.22108/IJGT.2021.130057.1735","DOIUrl":"https://doi.org/10.22108/IJGT.2021.130057.1735","url":null,"abstract":"We study the automorphism groups of finite-dimensional cyclic Leibniz algebras. In this connection, we consider the relationships between groups, modules over associative rings and Leibniz algebras.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46187140","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 : 2021-07-14DOI: 10.22108/IJGT.2021.128791.1696
O. Stoytchev
dWe exhibit presentations of the Von Dyck groups $D(2, 3, m), mge 3$, in terms of two generators of order $m$ satisfying three relations, one of which is Artin's braid relation. By dropping the relation which fixes the order of the generators we obtain the universal covering groups of the corresponding Von Dyck groups. In the cases $m=3, 4, 5$, these are respectively the double covers of the finite rotational tetrahedral, octahedral and icosahedral groups. When $mge 6$ we obtain infinite covers of the corresponding infinite Von Dyck groups. The interesting cases arise for $mge 7$ when these groups act as discrete groups of isometries of the hyperbolic plane. Imposing a suitable third relation we obtain three-relator presentations of $text{PSL}(2,m)$. We discover two general formulas presenting these as factors of $D(2, 3, m)$. The first one works for any odd $m$ and is essentially equivalent to the shortest known presentation of Sunday cite{Sunday}. The second applies to the cases $mequivpm 2 (text{mod} 3)$, $m ≢ 11(text{mod} 30)$, and is substantively shorter. Additionally, by random search, we find many efficient presentations of finite simple Chevalley groups PSL($2,q$) as factors of $D(2, 3, m)$ where $m$ divides the order of the group. The only other simple group that we found in this way is the sporadic Janko group $J_2$.
{"title":"On efficient presentations of the groups $text{PSL}(2,m)$","authors":"O. Stoytchev","doi":"10.22108/IJGT.2021.128791.1696","DOIUrl":"https://doi.org/10.22108/IJGT.2021.128791.1696","url":null,"abstract":"dWe exhibit presentations of the Von Dyck groups $D(2, 3, m), mge 3$, in terms of two generators of order $m$ satisfying three relations, one of which is Artin's braid relation. By dropping the relation which fixes the order of the generators we obtain the universal covering groups of the corresponding Von Dyck groups. In the cases $m=3, 4, 5$, these are respectively the double covers of the finite rotational tetrahedral, octahedral and icosahedral groups. When $mge 6$ we obtain infinite covers of the corresponding infinite Von Dyck groups. The interesting cases arise for $mge 7$ when these groups act as discrete groups of isometries of the hyperbolic plane. Imposing a suitable third relation we obtain three-relator presentations of $text{PSL}(2,m)$. We discover two general formulas presenting these as factors of $D(2, 3, m)$. The first one works for any odd $m$ and is essentially equivalent to the shortest known presentation of Sunday cite{Sunday}. The second applies to the cases $mequivpm 2 (text{mod} 3)$, $m ≢ 11(text{mod} 30)$, and is substantively shorter. Additionally, by random search, we find many efficient presentations of finite simple Chevalley groups PSL($2,q$) as factors of $D(2, 3, m)$ where $m$ divides the order of the group. The only other simple group that we found in this way is the sporadic Janko group $J_2$.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47396658","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 : 2021-07-10DOI: 10.22108/IJGT.2021.128396.1687
E. Bertram
P.Hall's classical equality for the number of conjugacy classes in p-groups yields k(G) >= (3/2)log_2 |G|when G is nilpotent. Using only Hall's theorem, this is the best one can do when |G| = 2^n. Using aresult of G.J. Sherman, we improve the constant 3/2 to 5/3, which is best possible across all nilpotentgroups and to 15/8 when G is nilpotent and |G| is not equal to 8 or 16. These results are then used to prove that k(G) > log_3 |G| when G/N is nilpotent, under natural conditions on N (normal in) G. Also,when G' is nilpotent of class c, we prove that k(G) >= (log |G|)^t when |G| is large enough, dependingonly on (c,t).
{"title":"New Lower Bounds for the Number of Conjugacy Classes in Finite Nilpotent Groups","authors":"E. Bertram","doi":"10.22108/IJGT.2021.128396.1687","DOIUrl":"https://doi.org/10.22108/IJGT.2021.128396.1687","url":null,"abstract":"P.Hall's classical equality for the number of conjugacy classes in p-groups yields k(G) >= (3/2)log_2 |G|when G is nilpotent. Using only Hall's theorem, this is the best one can do when |G| = 2^n. Using aresult of G.J. Sherman, we improve the constant 3/2 to 5/3, which is best possible across all nilpotentgroups and to 15/8 when G is nilpotent and |G| is not equal to 8 or 16. These results are then used to prove that k(G) > log_3 |G| when G/N is nilpotent, under natural conditions on N (normal in) G. Also,when G' is nilpotent of class c, we prove that k(G) >= (log |G|)^t when |G| is large enough, dependingonly on (c,t).","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68205313","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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