Pub Date : 2022-08-08DOI: 10.1142/S021819672350025X
S. D. Freedman, A. Lucchini, Daniele Nemmi, C. Roney-Dougal
We say that a finite group $G$ satisfies the independence property if, for every pair of distinct elements $x$ and $y$ of $G$, either ${x,y}$ is contained in a minimal generating set for $G$ or one of $x$ and $y$ is a power of the other. We give a complete classification of the finite groups with this property, and in particular prove that every such group is supersoluble. A key ingredient of our proof is a theorem showing that all but three finite almost simple groups $H$ contain an element $s$ such that the maximal subgroups of $H$ containing $s$, but not containing the socle of $H$, are pairwise non-conjugate.
{"title":"Finite groups satisfying the independence property","authors":"S. D. Freedman, A. Lucchini, Daniele Nemmi, C. Roney-Dougal","doi":"10.1142/S021819672350025X","DOIUrl":"https://doi.org/10.1142/S021819672350025X","url":null,"abstract":"We say that a finite group $G$ satisfies the independence property if, for every pair of distinct elements $x$ and $y$ of $G$, either ${x,y}$ is contained in a minimal generating set for $G$ or one of $x$ and $y$ is a power of the other. We give a complete classification of the finite groups with this property, and in particular prove that every such group is supersoluble. A key ingredient of our proof is a theorem showing that all but three finite almost simple groups $H$ contain an element $s$ such that the maximal subgroups of $H$ containing $s$, but not containing the socle of $H$, are pairwise non-conjugate.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82359967","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 : 2022-07-16DOI: 10.1142/s0218196723500030
Sneha Mavi, Anuj Bishnoi
In this paper, for a valued field $(K, v)$ of arbitrary rank and an extension $w$ of $v$ to $K(X),$ a relation between induced complete sequences of abstract key polynomials and MacLane-Vaqui'e chains is given.
{"title":"Abstract key polynomials and MacLane-Vaquié chains","authors":"Sneha Mavi, Anuj Bishnoi","doi":"10.1142/s0218196723500030","DOIUrl":"https://doi.org/10.1142/s0218196723500030","url":null,"abstract":"In this paper, for a valued field $(K, v)$ of arbitrary rank and an extension $w$ of $v$ to $K(X),$ a relation between induced complete sequences of abstract key polynomials and MacLane-Vaqui'e chains is given.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75908446","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 : 2022-06-09DOI: 10.1142/S0218196722500606
A. Panasenko
. We study prime and semiprime Novikov algebras. We prove that prime nonassociative Novikov algebra has zero nucleus and center. It is well known that an ideal of an alternative (semi)prime algebra is (semi)prime algebra. We proved this statement for Novikov algebras. It implies that a Baer radical exists in a class of Novikov algebras. Also, we proved that a minimal ideal of Novikov algebra is either trivial, or a simple algebra.
{"title":"Semiprime Novikov algebras","authors":"A. Panasenko","doi":"10.1142/S0218196722500606","DOIUrl":"https://doi.org/10.1142/S0218196722500606","url":null,"abstract":". We study prime and semiprime Novikov algebras. We prove that prime nonassociative Novikov algebra has zero nucleus and center. It is well known that an ideal of an alternative (semi)prime algebra is (semi)prime algebra. We proved this statement for Novikov algebras. It implies that a Baer radical exists in a class of Novikov algebras. Also, we proved that a minimal ideal of Novikov algebra is either trivial, or a simple algebra.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74599629","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 : 2022-06-05DOI: 10.1142/s0218196722500576
Z. Balogh
Let F G be the group algebra of a finite p -group G over a finite field F of positive characteristic p . Let ⊛ be an involution of the algebra F G which is a linear extension of an anti-automorphism of the group G to F G . If p is an odd prime, then the order of the ⊛ -unitary subgroup of F G is established. For the case p = 2 we generalize a result obtained for finite abelian 2-groups. It is proved that the order of the ∗ -unitary subgroup of F G of a non-abelian 2-group is always divisible by a number which depends only on the size of F , the order of G and the number of elements of order two in G . Moreover, we show that the order of the ∗ -unitary subgroup of F G determines the order of the finite p -group G .
{"title":"The order of the unitary subgroups of group algebras","authors":"Z. Balogh","doi":"10.1142/s0218196722500576","DOIUrl":"https://doi.org/10.1142/s0218196722500576","url":null,"abstract":"Let F G be the group algebra of a finite p -group G over a finite field F of positive characteristic p . Let ⊛ be an involution of the algebra F G which is a linear extension of an anti-automorphism of the group G to F G . If p is an odd prime, then the order of the ⊛ -unitary subgroup of F G is established. For the case p = 2 we generalize a result obtained for finite abelian 2-groups. It is proved that the order of the ∗ -unitary subgroup of F G of a non-abelian 2-group is always divisible by a number which depends only on the size of F , the order of G and the number of elements of order two in G . Moreover, we show that the order of the ∗ -unitary subgroup of F G determines the order of the finite p -group G .","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75930468","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 : 2022-05-20DOI: 10.1142/s0218196722500503
F. Benanti, A. Valenti
{"title":"*-Graded Capelli polynomials and their asymptotics","authors":"F. Benanti, A. Valenti","doi":"10.1142/s0218196722500503","DOIUrl":"https://doi.org/10.1142/s0218196722500503","url":null,"abstract":"","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77003569","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 : 2022-05-18DOI: 10.1142/s0218196722500424
M. Elhamdadi, A. Makhlouf, S. Silvestrov, E. Zappala
The purpose of this paper is to introduce and investigate the notion of derivation for quandle algebras. More precisely, we describe the symmetries on structure constants providing a characterization for a linear map to be a derivation. We obtain a complete characterization of derivations in the case of quandle algebras of dihedral quandles over fields of characteristic zero, and provide the dimensionality of the Lie algebra of derivations. Many explicit examples and computations are given over both zero and positive characteristic. Furthermore, we investigate inner derivations, in the sense of Schafer for non-associative structures. We obtain necessary conditions for the Lie transformation algebra of quandle algebras of Alexander quandles, with explicit computations in low dimensions.
{"title":"Derivation problem for quandle algebras","authors":"M. Elhamdadi, A. Makhlouf, S. Silvestrov, E. Zappala","doi":"10.1142/s0218196722500424","DOIUrl":"https://doi.org/10.1142/s0218196722500424","url":null,"abstract":"The purpose of this paper is to introduce and investigate the notion of derivation for quandle algebras. More precisely, we describe the symmetries on structure constants providing a characterization for a linear map to be a derivation. We obtain a complete characterization of derivations in the case of quandle algebras of dihedral quandles over fields of characteristic zero, and provide the dimensionality of the Lie algebra of derivations. Many explicit examples and computations are given over both zero and positive characteristic. Furthermore, we investigate inner derivations, in the sense of Schafer for non-associative structures. We obtain necessary conditions for the Lie transformation algebra of quandle algebras of Alexander quandles, with explicit computations in low dimensions.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82165378","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 : 2022-05-10DOI: 10.1142/s0218196722500497
Meng Gao, Wen Ting Zhang, Yanfeng Luo
{"title":"Finite basis problem for Catalan monoids with involution","authors":"Meng Gao, Wen Ting Zhang, Yanfeng Luo","doi":"10.1142/s0218196722500497","DOIUrl":"https://doi.org/10.1142/s0218196722500497","url":null,"abstract":"","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74049867","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 : 2022-05-09DOI: 10.1142/s0218196722500679
D. Burde, K. Dekimpe, Mina Monadjem
We study rigidity questions for pairs of Lie algebras $(mathfrak{g},mathfrak{n})$ admitting a post-Lie algebra structure. We show that if $mathfrak{g}$ is semisimple and $mathfrak{n}$ is arbitrary, then we have rigidity in the sense that $mathfrak{g}$ and $mathfrak{n}$ must be isomorphic. The proof uses a result on the decomposition of a Lie algebra $mathfrak{g}=mathfrak{s}_1dotplus mathfrak{s}_2$ as the direct vector space sum of two semisimple subalgebras. We show that $mathfrak{g}$ must be semisimple and hence isomorphic to the direct Lie algebra sum $mathfrak{g}cong mathfrak{s}_1oplus mathfrak{s}_2$. This solves some open existence questions for post-Lie algebra structures on pairs of Lie algebras $(mathfrak{g},mathfrak{n})$. We prove additional existence results for pairs $(mathfrak{g},mathfrak{n})$, where $mathfrak{g}$ is complete, and for pairs, where $mathfrak{g}$ is reductive with $1$-dimensional center and $mathfrak{n}$ is solvable or nilpotent.
{"title":"Rigidity results for Lie algebras admitting a post-Lie algebra structure","authors":"D. Burde, K. Dekimpe, Mina Monadjem","doi":"10.1142/s0218196722500679","DOIUrl":"https://doi.org/10.1142/s0218196722500679","url":null,"abstract":"We study rigidity questions for pairs of Lie algebras $(mathfrak{g},mathfrak{n})$ admitting a post-Lie algebra structure. We show that if $mathfrak{g}$ is semisimple and $mathfrak{n}$ is arbitrary, then we have rigidity in the sense that $mathfrak{g}$ and $mathfrak{n}$ must be isomorphic. The proof uses a result on the decomposition of a Lie algebra $mathfrak{g}=mathfrak{s}_1dotplus mathfrak{s}_2$ as the direct vector space sum of two semisimple subalgebras. We show that $mathfrak{g}$ must be semisimple and hence isomorphic to the direct Lie algebra sum $mathfrak{g}cong mathfrak{s}_1oplus mathfrak{s}_2$. This solves some open existence questions for post-Lie algebra structures on pairs of Lie algebras $(mathfrak{g},mathfrak{n})$. We prove additional existence results for pairs $(mathfrak{g},mathfrak{n})$, where $mathfrak{g}$ is complete, and for pairs, where $mathfrak{g}$ is reductive with $1$-dimensional center and $mathfrak{n}$ is solvable or nilpotent.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77674607","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 : 2022-05-06DOI: 10.1142/s0218196722500448
Nguyên Quang Lôc, N. Minh, P. Thuy
Let [Formula: see text] be a two-dimensional squarefree monomial ideal in a polynomial ring [Formula: see text], where [Formula: see text] is a field. In this paper, we give explicit formulas for the extremal Betti numbers of the [Formula: see text]th symbolic power of [Formula: see text] for all [Formula: see text]. As a consequence, we characterize the rings [Formula: see text] which are pseudo-Gorenstein as sense of Ene et al. [Pseudo-Gorenstein and level Hibi rings, J. Algebra 431 (2015) 138–161]. We also provide a complete classification for the level property of the second symbolic power [Formula: see text]. In particular, we obtain a new algebraic-property of the unknown Moore graph of degree 57.
{"title":"Extremal Betti numbers of symbolic powers of two-dimensional squarefree monomial ideals","authors":"Nguyên Quang Lôc, N. Minh, P. Thuy","doi":"10.1142/s0218196722500448","DOIUrl":"https://doi.org/10.1142/s0218196722500448","url":null,"abstract":"Let [Formula: see text] be a two-dimensional squarefree monomial ideal in a polynomial ring [Formula: see text], where [Formula: see text] is a field. In this paper, we give explicit formulas for the extremal Betti numbers of the [Formula: see text]th symbolic power of [Formula: see text] for all [Formula: see text]. As a consequence, we characterize the rings [Formula: see text] which are pseudo-Gorenstein as sense of Ene et al. [Pseudo-Gorenstein and level Hibi rings, J. Algebra 431 (2015) 138–161]. We also provide a complete classification for the level property of the second symbolic power [Formula: see text]. In particular, we obtain a new algebraic-property of the unknown Moore graph of degree 57.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81008473","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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