Let G be a finite group, and let Γ be a subset of G. The Kazhdan constant of the pair (G,Γ) is defined to bethe maximum distance we can guarantee that an arbitrary unitvector in an arbitrary nontrivial irreducible unitary representation space of G can be moved by some element of Γ. The Kazhdanconstant relates to the expansion properties of the Cayley graph generated by G and Γ, and has been much studied in this context. Different pairs (G1,Γ1) and (G2,Γ2) may give rise to isomorphic Cayley graphs. In this paper, we investigate the question: To whatextent is the Kazhdan constant a graph invariant? In other words, if the pairs yield isomorphic Cayley graphs, must the corresponding Kazhdan constants be equal? In our main theorem, we constructan infinite family of such pairs where the Kazhdan constants areunequal. Other relevant results are presented as well.
{"title":"Kazhdan constants and isomorphic graph pairs","authors":"M. Davila, Travis Hayes, Mike Krebs, Marcos Reyes","doi":"10.12958/adm1851","DOIUrl":"https://doi.org/10.12958/adm1851","url":null,"abstract":"Let G be a finite group, and let Γ be a subset of G. The Kazhdan constant of the pair (G,Γ) is defined to bethe maximum distance we can guarantee that an arbitrary unitvector in an arbitrary nontrivial irreducible unitary representation space of G can be moved by some element of Γ. The Kazhdanconstant relates to the expansion properties of the Cayley graph generated by G and Γ, and has been much studied in this context. Different pairs (G1,Γ1) and (G2,Γ2) may give rise to isomorphic Cayley graphs. In this paper, we investigate the question: To whatextent is the Kazhdan constant a graph invariant? In other words, if the pairs yield isomorphic Cayley graphs, must the corresponding Kazhdan constants be equal? In our main theorem, we constructan infinite family of such pairs where the Kazhdan constants areunequal. Other relevant results are presented as well.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66420267","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 Frobenius group G belongs to an important class of groups that more than 100 years ago was defined by F. G. Frobenius who proved that G is a semi-direct product of a normal subgroup K of G called kernel by another non-trivial subgroup H called the complement. In this case we show that a few of the classical finite groups can be Frobenius complement.
Frobenius群G属于100多年前由F. G. Frobenius定义的一类重要群，他证明了G是G的正规子群K(称为核)与另一个非平凡子群H(称为补)的半直积。在这种情况下，我们证明了一些经典有限群可以是Frobenius补。
{"title":"Classical groups as Frobenius complement","authors":"Mohammadreza Darefsheh, Hadiseh Saydi","doi":"10.12958/adm1929","DOIUrl":"https://doi.org/10.12958/adm1929","url":null,"abstract":"The Frobenius group G belongs to an important class of groups that more than 100 years ago was defined by F. G. Frobenius who proved that G is a semi-direct product of a normal subgroup K of G called kernel by another non-trivial subgroup H called the complement. In this case we show that a few of the classical finite groups can be Frobenius complement.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66420682","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 generalize to associative superalgebras Gerstenhaber's work on cohomology structure of an associative algebra. We introduce formal deformation theory of associative superalgebras.
{"title":"Cohomology and deformation of an associative superalgebra","authors":"R. Yadav","doi":"10.12958/adm2020","DOIUrl":"https://doi.org/10.12958/adm2020","url":null,"abstract":"In this paper we generalize to associative superalgebras Gerstenhaber's work on cohomology structure of an associative algebra. We introduce formal deformation theory of associative superalgebras.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66420808","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. Imam, S. Ibrahim, G. U. Garba, L. Usman, A. Idris
Let Xn be the finite set {1,2,3· · ·, n} and On defined by On={α∈Tn:(∀x, y∈Xn), x⩽y→xα⩽yα}be the semigroup of full order-preserving mapping on Xn. A transformation α in On is called quasi-idempotent if α=α2=α4. We characterise quasi-idempotent in On and show that the semigroup On is quasi-idempotent generated. Moreover, we obtained an upper bound forquasi-idempotents rank of On, that is, we showed that the cardinality of a minimum quasi-idempotents generating set for On is less than or equal to ⌈3(n−2)2⌉ where ⌈x⌉ denotes the least positive integerm such that x⩽m
{"title":"Quasi-idempotents in finite semigroup of full order-preserving transformations","authors":"A. Imam, S. Ibrahim, G. U. Garba, L. Usman, A. Idris","doi":"10.12958/adm1846","DOIUrl":"https://doi.org/10.12958/adm1846","url":null,"abstract":"Let Xn be the finite set {1,2,3· · ·, n} and On defined by On={α∈Tn:(∀x, y∈Xn), x⩽y→xα⩽yα}be the semigroup of full order-preserving mapping on Xn. A transformation α in On is called quasi-idempotent if α=α2=α4. We characterise quasi-idempotent in On and show that the semigroup On is quasi-idempotent generated. Moreover, we obtained an upper bound forquasi-idempotents rank of On, that is, we showed that the cardinality of a minimum quasi-idempotents generating set for On is less than or equal to ⌈3(n−2)2⌉ where ⌈x⌉ denotes the least positive integerm such that x⩽m","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66420218","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}
We study automorphisms of the semigroup BFZ with the family F of inductive nonempty subsets of ω and provethat the group Aut(BFZ) of automorphisms of the semigroup BFZ is isomorphic to the additive group of integers.
{"title":"On the group of automorphisms of the semigroup BFZ with the family F of inductive nonempty subsets of ω","authors":"O. Gutik, I. Pozdniakova","doi":"10.12958/adm2010","DOIUrl":"https://doi.org/10.12958/adm2010","url":null,"abstract":"We study automorphisms of the semigroup BFZ with the family F of inductive nonempty subsets of ω and provethat the group Aut(BFZ) of automorphisms of the semigroup BFZ is isomorphic to the additive group of integers.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"3 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66420732","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}
We consider the Munn matrix algebras over anassociative unitalK-algebraA, whereKis a commutative (unital)ring andAas aK-module is free (of őnite or inőnite rank), and,for each (not necessarily őnitely deőned) presentation ofA, we givepresentations of the Munn matrix algebras over it.
{"title":"Presentations of Munn matrix algebras over K-algebras with K being a commutative ring","authors":"V. Bondarenko","doi":"10.12958/adm2084","DOIUrl":"https://doi.org/10.12958/adm2084","url":null,"abstract":"We consider the Munn matrix algebras over anassociative unitalK-algebraA, whereKis a commutative (unital)ring andAas aK-module is free (of őnite or inőnite rank), and,for each (not necessarily őnitely deőned) presentation ofA, we givepresentations of the Munn matrix algebras over it.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66421071","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}
We show that the units found in torsion-free group rings by Gardam are twisted unitary elements. This justifies some choices in Gardam's construction that might have appeared arbitrary, and yields more examples of units. We note that all units found up to date exhibit non-trivial symmetry.
{"title":"On Gardam's and Murray's units in group rings","authors":"L. Bartholdi","doi":"10.12958/adm2053","DOIUrl":"https://doi.org/10.12958/adm2053","url":null,"abstract":"We show that the units found in torsion-free group rings by Gardam are twisted unitary elements. This justifies some choices in Gardam's construction that might have appeared arbitrary, and yields more examples of units. We note that all units found up to date exhibit non-trivial symmetry.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45643699","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 investigate q-Varchenko matrices for some hyperplane arrangements with symmetry in two andthree dimensions, and prove that they have a Smith normal formover Z[q]. In particular, we examine the hyperplane arrangement forthe regular n-gon in the plane and the dihedral model in the spaceand Platonic polyhedra. In each case, we prove that the q-Varchenko matrix associated with the hyperplane arrangement has a Smith normal form over Z[q] and realize their congruent transformation matrices over Z[q] as well.
{"title":"On Smith normal forms of q-Varchenko matrices","authors":"N. Boulware, N. Jing, Kailash C. Misra","doi":"10.12958/adm2006","DOIUrl":"https://doi.org/10.12958/adm2006","url":null,"abstract":"In this paper, we investigate q-Varchenko matrices for some hyperplane arrangements with symmetry in two andthree dimensions, and prove that they have a Smith normal formover Z[q]. In particular, we examine the hyperplane arrangement forthe regular n-gon in the plane and the dihedral model in the spaceand Platonic polyhedra. In each case, we prove that the q-Varchenko matrix associated with the hyperplane arrangement has a Smith normal form over Z[q] and realize their congruent transformation matrices over Z[q] as well.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46806524","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 celebrated result of Herstein [10, Theorem 6] states that a ring R must be commutative if[x,y]n(x,y)=[x,y] for all x, y ∈ R, wheren (x,y)>1 is an integer. In this paper, we investigate the structure of a prime ring satisfies the identity F([x,y])n=F([x,y]) and σ([x,y])n=σ([x,y]), where F and σ are generalized derivation and automorphism of a prime ring R, respectively and n>1a fixed integer.
{"title":"On Herstein's identity in prime rings","authors":"G. Sandhu","doi":"10.12958/adm1581","DOIUrl":"https://doi.org/10.12958/adm1581","url":null,"abstract":"A celebrated result of Herstein [10, Theorem 6] states that a ring R must be commutative if[x,y]n(x,y)=[x,y] for all x, y ∈ R, wheren (x,y)>1 is an integer. In this paper, we investigate the structure of a prime ring satisfies the identity F([x,y])n=F([x,y]) and σ([x,y])n=σ([x,y]), where F and σ are generalized derivation and automorphism of a prime ring R, respectively and n>1a fixed integer.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66418591","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 L be an algebra over a field F with the binary operations + and [·,·]. Then L is called a left Leibniz algebra if [[a,b],c]=[a,[b,c]]−[b,[a,c]] for all a, b, c ∈ L. We describe the inner structure of left Leibniz algebras having dimension 3.
{"title":"On the structure of low-dimensional Leibniz algebras: some revision","authors":"L. A. Kurdachenko, O. Pypka, I. Subbotin","doi":"10.12958/adm2036","DOIUrl":"https://doi.org/10.12958/adm2036","url":null,"abstract":"Let L be an algebra over a field F with the binary operations + and [·,·]. Then L is called a left Leibniz algebra if [[a,b],c]=[a,[b,c]]−[b,[a,c]] for all a, b, c ∈ L. We describe the inner structure of left Leibniz algebras having dimension 3.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66420878","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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