We consider elements which are both of high multiplicative order and normal in extensions (F_{q^{m} } ) of the field (F_{q} ). If the extension is defined by a cyclotomic polynomial, we construct such elements explicitly and give explicit lower bounds on their orders.
{"title":"Normal high order elements in finite field extensions based on the cyclotomic polynomials","authors":"R. Popovych, Ruslan Skuratovskii","doi":"10.12958/adm1117","DOIUrl":"https://doi.org/10.12958/adm1117","url":null,"abstract":"We consider elements which are both of high multiplicative order and normal in extensions (F_{q^{m} } ) of the field (F_{q} ). If the extension is defined by a cyclotomic polynomial, we construct such elements explicitly and give explicit lower bounds on their orders.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43910863","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}
H. Ansari-Toroghy, Faranak Farshadifar, S. Maleki-Roudposhti
Let (R) be a commutative ring with identity. A proper submodule (N) of an (R)-module (M) is said to be a 2-absorbing submodule of (M) if whenever (abm in N) for some (a, b in R) and (m in M), then (am in N) or (bm in N) or (ab in (N :_R M)). In [3], the authors introduced two dual notion of 2-absorbing submodules (that is, 2-absorbing and strongly 2-absorbing second submodules) of (M) and investigated some properties of these classes of modules. In this paper, we will introduce the concepts of generalized 2-absorbing and strongly generalized 2-absorbing second submodules of modules over a commutative ring and obtain some related results.
{"title":"Generalized 2-absorbing and strongly generalized 2-absorbing second submodules","authors":"H. Ansari-Toroghy, Faranak Farshadifar, S. Maleki-Roudposhti","doi":"10.12958/adm585","DOIUrl":"https://doi.org/10.12958/adm585","url":null,"abstract":"Let (R) be a commutative ring with identity. A proper submodule (N) of an (R)-module (M) is said to be a 2-absorbing submodule of (M) if whenever (abm in N) for some (a, b in R) and (m in M), then (am in N) or (bm in N) or (ab in (N :_R M)). In [3], the authors introduced two dual notion of 2-absorbing submodules (that is, 2-absorbing and strongly 2-absorbing second submodules) of (M) and investigated some properties of these classes of modules. In this paper, we will introduce the concepts of generalized 2-absorbing and strongly generalized 2-absorbing second submodules of modules over a commutative ring and obtain some related results.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49476366","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 [1] the author gives a description of Poisson brackets on some algebras of quantum polynomials (mathcal{O}_q), which is called the general algebra of quantum polynomials. The main of this paper is to present a generalization of [1] through a description of Poisson brackets on some skew PBW extensions of a ring (A) by the extensions (mathcal{O}_{q,delta}^{r,n}), which are generalization of (mathcal{O}_q), and show some examples of skew PBW extension where we can apply this description.
{"title":"Poisson brackets on some skew PBW extensions","authors":"Brian Andres Zambrano Luna","doi":"10.12958/adm1037","DOIUrl":"https://doi.org/10.12958/adm1037","url":null,"abstract":"In [1] the author gives a description of Poisson brackets on some algebras of quantum polynomials (mathcal{O}_q), which is called the general algebra of quantum polynomials. The main of this paper is to present a generalization of [1] through a description of Poisson brackets on some skew PBW extensions of a ring (A) by the extensions (mathcal{O}_{q,delta}^{r,n}), which are generalization of (mathcal{O}_q), and show some examples of skew PBW extension where we can apply this description.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44122429","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 the paper, we consider a two-symbol system of encoding for real numbers with two bases having different signs ({g_0<1}) and (g_1=g_0-1). Transformations (bijections of the set to itself) of interval ([0,g_0]) preserving tails of this representation of numbers are studied. We prove constructively that the set of all continuous transformations from this class with respect to composition of functions forms an infinite non-abelian group such that increasing transformations form its proper subgroup. This group is a proper subgroup of the group of transformations preserving frequencies of digits of representations of numbers.
{"title":"Group of continuous transformations of real interval preserving tails of G2-representation of numbers","authors":"M. Pratsiovytyi, Iryna Lysenko, Yuliya Maslova","doi":"10.12958/adm1498","DOIUrl":"https://doi.org/10.12958/adm1498","url":null,"abstract":"In the paper, we consider a two-symbol system of encoding for real numbers with two bases having different signs ({g_0<1}) and (g_1=g_0-1). Transformations (bijections of the set to itself) of interval ([0,g_0]) preserving tails of this representation of numbers are studied. We prove constructively that the set of all continuous transformations from this class with respect to composition of functions forms an infinite non-abelian group such that increasing transformations form its proper subgroup. This group is a proper subgroup of the group of transformations preserving frequencies of digits of representations of numbers.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42277834","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 obtain the description of non-periodic locally generalized radical groups whose cyclic subgroups are (GNA)-subgroups.
本文给出了循环子群为（GNA）-子群的非周期局部广义根群的描述。
{"title":"On some non-periodic groups whose cyclic subgroups are GNA-subgroups","authors":"A. A. Pypka","doi":"10.12958/adm548","DOIUrl":"https://doi.org/10.12958/adm548","url":null,"abstract":"In this paper we obtain the description of non-periodic locally generalized radical groups whose cyclic subgroups are (GNA)-subgroups.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48799971","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 author studies groups with given restrictions on norms of decomposable and Abelian non-cyclic subgroups. The properties of non-periodic locally soluble groups, in which such norms are nonidentity and have the identity intersection, are described.
{"title":"Locally soluble groups with the restrictions on the generalized norms","authors":"T. Lukashova","doi":"10.12958/adm1527","DOIUrl":"https://doi.org/10.12958/adm1527","url":null,"abstract":"The author studies groups with given restrictions on norms of decomposable and Abelian non-cyclic subgroups. The properties of non-periodic locally soluble groups, in which such norms are nonidentity and have the identity intersection, are described.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42637462","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}
Let (G) be a finite group and (P) be a (p)-subgroup of (G). If (P) is a Sylow subgroup of some normal subgroup of (G), then we say that (P) is normally embedded in (G). Groups with normally embedded maximal subgroups of Sylow (p)-subgroup, where ({(|G|, p-1)=1}), are studied. In particular, the (p)-nilpotency of such groups is proved.
{"title":"On p-nilpotency of finite group with normally embedded maximal subgroups of some Sylow subgroups","authors":"A. Trofimuk","doi":"10.12958/adm1128","DOIUrl":"https://doi.org/10.12958/adm1128","url":null,"abstract":"Let (G) be a finite group and (P) be a (p)-subgroup of (G). If (P) is a Sylow subgroup of some normal subgroup of (G), then we say that (P) is normally embedded in (G). Groups with normally embedded maximal subgroups of Sylow (p)-subgroup, where ({(|G|, p-1)=1}), are studied. In particular, the (p)-nilpotency of such groups is proved.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44880222","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 subset (X) of prime power order elements of a finite group (G) is called pp-independent if there is no proper subset (Y) of (X) such that (langle Y,Phi(G) rangle = langle X,Phi(G) rangle), where (Phi(G)) is the Frattini subgroup of (G). A group (G) has property (mathcal{B}_{pp}) if all pp-independent generating sets of (G) have the same size. (G) has the pp-basis exchange property if for any pp-independent generating sets (B_1, B_2) of (G) and (xin B_1) there exists (yin B_2) such that ((B_1setminus {x})cup {y}) is a pp-independent generating set of (G). In this paper we describe all finite solvable groups with property (mathcal{B}_{pp}) and all finite solvable groups with the pp-basis exchange property.
{"title":"Sets of prime power order generators of finite groups","authors":"A. Stocka","doi":"10.12958/adm1479","DOIUrl":"https://doi.org/10.12958/adm1479","url":null,"abstract":"A subset (X) of prime power order elements of a finite group (G) is called pp-independent if there is no proper subset (Y) of (X) such that (langle Y,Phi(G) rangle = langle X,Phi(G) rangle), where (Phi(G)) is the Frattini subgroup of (G). A group (G) has property (mathcal{B}_{pp}) if all pp-independent generating sets of (G) have the same size. (G) has the pp-basis exchange property if for any pp-independent generating sets (B_1, B_2) of (G) and (xin B_1) there exists (yin B_2) such that ((B_1setminus {x})cup {y}) is a pp-independent generating set of (G). In this paper we describe all finite solvable groups with property (mathcal{B}_{pp}) and all finite solvable groups with the pp-basis exchange property.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48767669","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}
L. Bartholdi, José Manuel Rodríguez Caballero, Ahmed Tanbir
We introduce self-similar algebras and groups closely related to the Thue-Morse sequence, and begin their investigation by describing a~character on them, the `spread' character.
{"title":"The Thue-Morse substitutions and self-similar groups and algebras","authors":"L. Bartholdi, José Manuel Rodríguez Caballero, Ahmed Tanbir","doi":"10.12958/adm1597","DOIUrl":"https://doi.org/10.12958/adm1597","url":null,"abstract":"We introduce self-similar algebras and groups closely related to the Thue-Morse sequence, and begin their investigation by describing a~character on them, the `spread' character.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44060168","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 note, we give a precise construction ofone of the families of 2-designs arose from studying ŕag-transitive 2-designs with parameters(v, k, λ) whose replication numbersrare coprime to λ. We show that for a given positive integer q=22n+1⩾8, there exists a 2-design with parameters (q2+ 1, q, q−1) and the replication numberq 2 admitting the Suzuki group Sz(q) asits automorphism group. We also construct a family of 2-designs with parameters (q2+ 1, q(q−1),(q−1)(q2−q−1)) and thereplication number q2(q−1) admitting the Suzuki groups Sz(q) astheir automorphism groups.
在本文中，我们给出了2-设计族的一个精确构造，该族是通过研究具有参数(v, k， λ)的ŕag-transitive 2-设计族而产生的，这些2-设计族的复制数很少质数为λ。我们表明，对于给定的正整数q=22n+1大于或等于8，存在具有参数(q2+ 1, q, q−1)的2-设计，并且复制数q2承认铃木组Sz(q)是其自同构组。我们还构造了一个具有参数(q2+ 1, q(q−1)，(q−1)(q2−q−1)和重复数q2(q−1)的2-设计族，允许Suzuki群Sz(q)为它们的自同构群。
{"title":"A note on two families of 2-designs arose from Suzuki-Tits ovoid","authors":"S. H. Alavi","doi":"10.12958/adm1687","DOIUrl":"https://doi.org/10.12958/adm1687","url":null,"abstract":"In this note, we give a precise construction ofone of the families of 2-designs arose from studying ŕag-transitive 2-designs with parameters(v, k, λ) whose replication numbersrare coprime to λ. We show that for a given positive integer q=22n+1⩾8, there exists a 2-design with parameters (q2+ 1, q, q−1) and the replication numberq 2 admitting the Suzuki group Sz(q) asits automorphism group. We also construct a family of 2-designs with parameters (q2+ 1, q(q−1),(q−1)(q2−q−1)) and thereplication number q2(q−1) admitting the Suzuki groups Sz(q) astheir automorphism groups.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43753148","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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