Publication Date: 1998
Acta Mathematica Hungarica (15882632)81(1-2)pp. 109-119
Let A be a non-zero Artinian R-module. For an arbitrary ideal I of R, we show that the local homology module Hpx(A) is independent of the choice of x whenever 0:A I = 0:A(x1,..., xr). Thus, identifying these modules, we write HpI(A). In this paper we prove that there is a certain duality between HiI(A) and the local cohomology modules and provide some information about the vanishing local homology module HiI(A) which gives an improved form of the main results of [22].
Publication Date: 1999
Archiv der Mathematik (0003889X)73(2)pp. 104-108
In this note we prove that every infinite group G is 3-abelian (i.e. (ab)3 = a3b3 for all a, b in G) if and only if in every two infinite subsets X and Y of G there exist x ∈ X and y ∈ Y such that (xy)3 = x3y3.
Publication Date: 1999
Communications in Algebra (00927872)27(11)pp. 5633-5638
In this note we show that if G is a finitely generated soluble group, then every infinite subset of G contains two elements generating a nilpotent group of class at most k if and only if G is finite by a group in which every two generator subgroup is nilpotent of class at most k.
Publication Date: 1999
Journal of Algebra (00218693)221(2)pp. 570-578
Let n be an integer greater than 1. A group G is said to be n-permutable whenever for every n-tuple (x1,...,xn) of elements of G there exists a non-identity permutation σ of {1,...,n} such that x1···xn=xσ(1)···xσ(n). In this paper we prove that an infinite group G is n-permutable if and only if for every n infinite subsets X1,...,Xn of G there exists a non-identity permutation σ on {1,...,n} such that X1···Xn∪Xσ(1)···Xσ(n)≠∅. © 1999 Academic Press.
Publication Date: 1999
Journal of Algebra (00218693)214(2)pp. 571-624
In this paper we study the Weyl groups of reduced extended affine root systems, the root systems of extended affine Lie algebras. We start by describing the extended affine Weyl group as a semidirect product of a finite Weyl group and a Heisenberg-like normal subgroup. This provides a unique expression for the Weyl group elements (in terms of some naturally arisen transformations) which is crucial in the further study of extended affine Weyl groups. We use this to give a presentation, called a presentation by conjugation, for an important subclass of extended affine Weyl groups. Using a new notion, called the index which is an invariant of the extended affine root systems, we show that one of the important features of finite and affine root systems (related to Weyl group) holds for the class of extended affine root systems. © 1999 Academic Press.
Publication Date: 1999
Kyoto Journal of Mathematics (0023608X)39(4)pp. 607-618
Publication Date: 1999
Communications in Algebra (00927872)27(12)pp. 6191-6198
Publication Date: 1999
Journal of Algebra (00218693)222(1)pp. 174-189
Extended affine Weyl groups are the Weyl groups of root systems of a new class of Lie algebras called extended affine Lie algebras. In this paper we show that a (reduced) extended affine Weyl group is the homomorphic image of some indefinite Kac-Moody Weyl group where the homomorphism and its kernel are given explicitly. © 1999 Academic Press.
Publication Date: 2000
Rendiconti del Seminario Matematico dell 'Universita' di Padova/Mathematical Journal of the University of Padova (22402926)104pp. 129-134
Let α1,…, αn be nonzero integers whose greatest common divisor is d. We prove that an infinite group G is of finite exponent dividing d if and only if for every n infinite subsets X1, …, Xn of G there exist x1 ∈ X1, …, xn ∈ Xn such that x1α1…xαnn = 1. © Rendiconti del Seminario Matematico della Università di Padova, 2000, tous droits réservés.
Publication Date: 2000
Communications in Algebra (00927872)28(1)pp. 465-488
In 1985 K. Saito [Sa1] introduced the concept of an extended affine Weyl group (EAWG), the Weyl group of an extended affine root system (EARS). In [A2, Section 5], we gave a presentation called "a presentation by conjugation" for the class of EAWGs of index zero, a subclass of EAWGs. In this paper we will givo a presentation which we call a "generalized presentation by conjugation" for the class of reduced EAWGs. If the extended affine Weyl group is of index zero this presentation reduces to "a presentation by conjugation". Our main result states that when the nullity of the EARS is 2, these two presentations coincide that is, EAWGs of nullity 2 have "a presentation by conjugation". In [ST] another presentation for EAWGs of nullity 2 is given. Copyright © 2000 by Marcel Dekker, Inc.
Publication Date: 2000
Communications in Algebra (00927872)28(6)pp. 2753-2781
We first give a characterization of the core (modulo its center) of an extended affine Lie algebra and then use this characterization to show that as in the case of affine Kac-Moody Lie algebras, many of the known examples of EALAs can be constructed from standard examples by a process known as "twisting".
Publication Date: 2000
Rendiconti del Seminario Matematico dell 'Universita' di Padova/Mathematical Journal of the University of Padova (22402926)103pp. 47-49
In this note, we prove that, in every finitely generated soluble group G, G/Z2 (G) is finite if and only if in every infinite subset X of G there exist different x, y such that [x, y, y] = 1. © Rendiconti del Seminario Matematico della Università di Padova, 2000, tous droits réservés.
Publication Date: 2000
Bulletin of the Australian Mathematical Society (00049727)62(1)pp. 141-148
Let k be a positive integer. We denote by εk(∞) the class of all groups in which every infinite subset contains two distinct elements cursive Greek chi, y such that [cursive Greek chi,k y] = 1. We say that a group G is an ε*k-group provided that whenever X,Y are infinite subsets of G, there exists cursive Greek chi ∈ X, y ∈ Y such that [cursive Greek chi,k y] = 1. Here we prove that: (1) If G is a finitely generated soluble group, then G ∈ ε3(∞) if and only if G is finite by a nilpotent group in which every two generator subgroup is nilpotent of class at most 3. (2) If G is a finitely generated metabelian group, then G ∈ εk(∞) if and only if G/Zk(G) is finite, where Zk(G) is the (k + 1)-th term of the upper central series of G. (3) If G is a finitely generated soluble εk(∞)-group, then there exists a positive integer t depending only on k such that G/Zt(G) is finite. (4) If G is an infinite ε*k-group in which every non-trivial finitely generated subgroup has a non-trivial finite quotient, then G is k-Engel. In particular, G is locally nilpotent.
Publication Date: 2001
Bulletin of the Australian Mathematical Society (00049727)64(1)pp. 27-31
We use Ramsey's theorem to generalise a result of L. Babai and T.S. Sós on Sidon subsets and then use this to prove that for an integer n > 1 the class of groups in which every infinite subset contains a rewritable n-subset coincides with the class of groups in which every infinite subset contains n mutually disjoint non-empty subsets X1, ..., Xn such that X1 ⋯ Xn ∩ Xσ(1) ⋯ Xσ(n) ≠ 0 for some non-identity permutation σ on the set {1, ..., n}.
Publication Date: 2001
Communications in Algebra (00927872)29(4)pp. 1571-1581
Let n > 1 be an integer. A group G is said to be n-rewritable, whenever for any subset {x1, . . ., xn} of elements of G, there exist distinct permutations τ, σ of the set {1, 2, . . ., n} such that xτ(1) · · · xτ(n) = xσ(1) · · · xσ(n). In this paper we show that an infinite group G is n-rewritable if and only if for every n infinite subsets X1, . . ., Xn of G there exist distinct permutations τ, σ of the set {1, 2, . . ., n} such that Xτ(1) · · · Xτ(n) ∩ Xσ(n) · · · Xσ(n) ≠ 0.
Publication Date: 2001
Colloquium Mathematicum (00101354)87(1)pp. 129-136
Let A be a Noetherian ring, let M be a finitely generated A-module and let φ be a system of ideals of A. We prove that, for any ideal a in φ, if, for every prime ideal p of A, there exists an integer k(p), depending on p, such that ak(p) kills the general local cohomology module Hjφ (Mp) for every integer j less than a fixed integer n, where φp := {ap : a ε φ}, then there exists an integer k such that akHjφ(M) = 0 for every j < n. © 2001, Instytut Matematyczny. All rights reserved.
Publication Date: 2001
Algebra Colloquium (02191733)8(2)pp. 153-157
Let α1,... ,αn ∈ ℕ. We prove that, in every infinite ring R, x1α1 ⋯xnαn = 0 for all x1,..., xn ∈ R if and only if, for any n infinite subsets X1,..., Xn of R, there exist x1 ∈ X1, ..., xn ∈ Xn such that x1α1 ⋯ xnαn = 0. © Inst. Math. CAS 2001.
Publication Date: 2001
Houston Journal of Mathematics (03621588)27(3)pp. 511-522
Let n and k be positive integers. We say that a group G satisfies the condition ε(n) (respectively, εk(n)) if and only if any set with n + 1 elements of G contains two distinct elements x, y such that [x,t y] = 1 for some positive integer t = t(x,y) (respectively, [x,k y] = 1). Here we study certain groups satisfying these conditions. We prove that if G is a finite group satisfying the condition ε(n), then G is nilpotent if n < 3 and G is soluble if n < 16. If G is a finitely generated soluble group satisfying the condition ε(2), then G is nilpotent. If k and n are positive integers and G is a finitely generated residually finite group satisfying the condition εk(n), then G is nilpotent if n < 3 and G is polycyclic if n < 16. In particular, there is a positive integer c depending only on k such that G/Zc(G) is finite, where Zc(G) is the (c + 1)-th term of the upper central series of G. Also these bounds cannot be improved.
Publication Date: 2002
Communications in Algebra (00927872)30(10)pp. 4821-4826
Publication Date: 2002
Journal of Lie Theory (09495932)12(2)pp. 515-527
There are two notions of the extended affine root systems in the literature which both are introduced axiomatically. One, extended affine root system (SAERS for short), consists only of nonisotropic roots, while the other, extended affine root system (EARS for short), contains certain isotropic roots too. We show that there is a one to one correspondence between (reduced) SEARSs and EARSs. Namely the set of nonisotropic roots of any EARS is a (reduced) SEARS, and conversely, there is a unique way of adding certain isotropic elements to a SEARS to get an EARS. (It is known that some of extended affine root systems are not the root system of any extended affine Lie algebra.).
Publication Date: 2002
Proceedings of the American Mathematical Society (10886826)130(10)pp. 2827-2836
For a given positive integer n and a given prime number p, let r = r (n,p) be the integer satisfying pr-1 < n ≤ pr. We show that every locally finite p-group, satisfying the n-Engel identity, is (nilpotent of n-bounded class)-by-(finite exponent) where the best upper bound for the exponent is either pr or pr-1 if p is odd. When p = 2 the best upper bound is pr-1 pr or pr+1. In the second part of the paper we focus our attention on 4-Engel groups. With the aid of the results of the first part we show that every 4-Engel 3-group is soluble and the derived length is bounded by some constant.
Publication Date: 2002
Communications in Algebra (00927872)30(2)pp. 859-867
Publication Date: 2002
Communications in Algebra (00927872)30(8)pp. 3813-3823
Publication Date: 2002
Bulletin of the Belgian Mathematical Society - Simon Stevin (13701444)9(2)pp. 205-215
In this paper, we consider some combinatorial conditions on infinite subsets of groups, and we obtain in terms of these conditions some characterizations of nilpotent-by-finite and finite-by-nilpotent groups on the class of finitely generated soluble groups.
Publication Date: 2003
Journal of the Australian Mathematical Society (14467887)75(3)pp. 313-324
Let R be a commutative Noetherian ring with nonzero identity and let M be a finitely generated R-module. In this paper, we prove that if an ideal I of R is generated by a u.s.d-sequence on M then the local cohomology module H Ii(M) is I-cofinite. Furthermore, for any system of ideals φ of R, we study the cofiniteness problem in the context of general local cohomology modules.
Publication Date: 2003
Bulletin of the Australian Mathematical Society (00049727)67(1)pp. 115-119
Let c ≥ 0, d ≥ 2 be integers and Nc(d) be the variety of groups in which every d-generator subgroup is nilpotent of class at most c. N.D. Gupta asked for what values of c and d is it true that Nc(d) is locally nilpotent? We prove that if c ≤ 2d + 2d-1 - 3 then the variety Nc(d) is locally nilpotent and we reduce the question of Gupta about the periodic groups in Nc(d) to the prime power exponent groups in this variety.
