DOCSLIB.ORG

Combinatorics and Parallel Structures in Torsion-Free Abelian Group Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Combinatorics and Parallel Structures in Torsion-Free Abelian Group Theory Combinatorics and parallel structures in torsion-free abelian group theory Ekaterina Blagoveshchenskaya St. Petersburg, Russia 1 Let X be a torsion-free abelian group of finite rank n. Let {a1,...,an} be a linearly independent system of its element ∼ X and let τi ⊂ Q, with Z ⊂ τi = ai∗ . Then A = τ1a1 ⊕ ... ⊕ τnan ⊂ X and for any b ∈ X \ A there exist integers α, αi such that αb = α1a1 + ... + αnan with at least two non-zero coefficients αi. Let B =∼ X/A. We may consider a torsion-free abelian group X of finite rank as an essential extension of a completely decomposable torsion-free group A by means of a torsion group B. By Definition, Tcr(A)= {τ1,...,τk} is called the set of critical types of A (repetitions are excluded, k ≤ n). If B is finite then A is uniquely determined up to isomorphism and Tcr(A)= Tcr(X). Moreover, there exists just one ”special” fully invariant subgroup of X isomorphic to A, which is called the regulator of X and denoted by R(X). Assume A = R(X). 2 Main Definitions A torsion-free abelian group is completely decomposable, a CD-group, if it is a direct sum of subgroups of the rationals Q. An ACD-group X (almost completely decomposable group) is a torsion-free abelian group of finite rank, that contains a fully invariant completely decomposable group A = R(X), its regulator, for which X/A is a finite group. In case of cyclic X/A we say that ACD-group X is a CRQ-group (with cyclic regulator quotient). 3 A.G. Kurosh. Theory of groups, M. Nauka, 1967. L. Fuchs. Infinite Abelian Groups, vol. 1, 2, Academic Press 1970, 1973. D. Arnold. Finite Rank Torsion Free Abelian Groups and Rings, Lecture Notes in Mathematics, vol. 931, Springer Verlag, 1982. A. Mader. Almost completely decomposable abelian groups, Gordon and Breach, Algebra, Logic and Applications Vol. 13, Amsterdam, 2000. P.Krylov, A. Michalev, A. Tuganbaev. Endomorphism rings of abelian groups, Kluwer Academic Publishes, Dordrecht-Boston-London, 2003. E. Blagoveshchenskaya. Almost Completely Decomposable Abelian Groups and their Endomorphism Rings, Mathematics in Polytechnical University, St. Petersburg, 2009. 4 X ∈ ACD: 1) End X ⊂ End A with A = Aτ and rk Aτ ≥ 1. Lτ∈Tcr(X) 2) X/A = Lp∈P Tp, ∼ 3) X = Pp∈P Xp, X/A = Pp∈P Xp/A = Pp∈P Tp with fully invariant, pure subgroups Xp with Xp/A =∼ Tp, 4) End Xp ⊂ End A and End X = Tp∈P End Xp. More precisely, X Xp = A +( M Aτ )∗(p) τ∈[Tp] with [Tp] the minimal subset of Tcr(X) such that X ( Aτ ) / Aτ =∼ Tp. Lτ∈[Tp] ∗(p) Lτ∈[Tp] 5 Definition Let G and H be torsion-free abelian groups of finite rank. Then G and H are called nearly isomorphic (in symbols G =∼nr H) if and only if for any prime q there are monomorphisms ηq : G −→ H and ξq : H −→ G such that H/ηq(G) and G/ξq(H) are finite groups and |H/ηq(G)| and q as well as |G/ξq(H)| and q are relatively prime. Arnold (1982): If G =∼nr H and G = G1 ⊕ ... ⊕ Gk then there exists a decomposition H = H1 ⊕ ... ⊕ Hk such that G1 =∼nr H1, ... , Gk =∼nr Hk. 6 This leads to near direct decomposition theory of CRQ-groups with pairwise incomparable critical types (Blagoveshchenskaya and Mader, 1994): If X = X1 ⊕ X2 then A = A1 ⊕ A2 with A1 ⊂ X1, A2 ⊂ X2. This implies X/A =∼ X1/A1 ⊕ X2/A2 and mτ (X)= mτ (X1)mτ (X2) for any τ. Near Baer-Kaplansky Theorem (Blagoveshchenskaya, Ivanov, Schultz (2001)): Let X,Y ∈ CRQ. Then End(X) =∼ End(Y ) if and only if X =∼nr Y . Question: What groups of infinite rank admit the similar decomposition theory up to the new (extending from CRQ-class) near-isomorphism equivalence? 7 Near-isomorphism for torsion-free abelian groups is expected: (1) to preserve decomposability properties of groups (in the sense of Arnold’s theorem) (2) to be isomorphism for completely decomposable groups (3) to be traditional near-isomorphism for groups of finite rank 8 Definition (Blagoveshchenskaya, Str¨ungmann, G¨obel, 2002 – 2007) Let G and H be torsion-free abelian groups. Then G and H are called nearly isomorphic, G =∼nr H if for every integer N there exist monomorphisms ϕN : G → H and ψN : H → G such that 1. H/GϕN and G/HψN are torsion; 2. (H/GϕN )p =0=(G/HψN )p for all primes p dividing N; 3. for every finite rank pure subgroups G′ ⊆ G and H′ ⊆ H the ′ H ′ ′ G ′ quotients (G ϕN )∗ /G ϕN and (H ψN )∗ /H ψN are finite. 9 Blagoveshchenskaya, G¨obel (2002): consideration of unions of ascending chains of finite rank crq-groups leads to Definition 1 Let T be a set of finitely or countably many pairwise incomparable ring types and suppose that A = Lτ∈T Aτ is a direct sum of τ-homogeneous cd-groups Aτ of finite rank. Then a torsion-free group X is called a local crq-group (X ∈ CRQloc) if the following conditions hold: (1) A ⊆ X (2) X/A is a direct sum of p-primary cyclic groups Tp of orders αp ep = p for distinct primes p from a set P and αp ∈ N. (3) Each Aτ , τ ∈ T = Tcr(X), is pure in X. Definition 2 Let X ∈ CRQloc. We call X an almost rigid group if it is finitely presented, that is each τ, τ ∈ T = Tcr(X), belongs to finitely many [Tp], p ∈ P . 10 New Definitions A group A0 is called a strongly decomposable group (sd-group) if it is a direct sum of strongly indecomposable groups. If a group B contains a strongly decomposable group as a subgroup of finite index, then B will be called an almost strongly decomposable group (asd-group). 11 L. Fuchs, C. Metelli, D. Arnold, C. Vinsonhailer (1991-1993): Let A = Aτ with Aτ = τaτ be a rigid completely Lτ∈Tcr(A) decomposable group of finite rank and let {ατ : τ ∈ Tcr(A)} be integers, for which the following hold, 1. rk A = |Tcr(A)|≥ 3, 2. for any τ ∈ Tcr(A) there is a prime p with τ(p)= ∞ and σ(p) = ∞ for all σ = τ, σ ∈ Tcr(A), Z 3. Tτ= σ τ = for each σ ∈ Tcr(A), 4. gcd({ατ | τ = σ, τ ∈ Tcr(A)}) = 1 for any σ ∈ Tcr(A), 5. each ατ is not p-divisible if σ(p)= ∞ for some σ ∈ Tcr(A). Let K(A) =∼ Z be a group generated by the element ατ aτ . Then the strongly indecomposable group Pτ∈Tcr(A) B(A)= A/K(A) is called a proper B(1)-group. 12 Definition (Blagoveshchenskaya, 2007) Let X be a rigid crq-group of ring type and let ατ , τ ∈ T , be integers such that the following conditions hold, 1) A = A ⊕ ... ⊕ At is the regulator of X with Ai = τaτ , 1 Lτ∈Ti Tcr(X)= Si=1,...,t Ti, Ti = Tcr(Ai) and Ti ∩ Tj = ∅ if i = j, 2) Let K = K(Ai) ⊂ X with K(Ai)= ατ aτ and Li≤t Pτ∈Ti φ : A −→ A/K. 3) Xpφ =∼ Xp and Xp and Xpφ are fully invariant, pure, p-generated subgroups over A = R(X) and A/K accordingly. The group B = X/K is called a B(1)crq-group. φ : X −→ B = X/K with a rigid crq-group X will be called a regular representation of a B(1)crq-group B. 13 Recall CRQ ⊂ ACD. ⊂ ⊂ CRQ −→ CRQloc −→ ACDloc ↓ (φ) ↓ (φ) ↓ (φ) (1) ⊂ (1) ⊂ (1) B CRQ −→ B CRQloc −→ B ASDloc Theorem (Blagoveshchenskaya, Str¨ungmann, G¨obel, 2011) Let X and Y be nearly isomorphic rigid ACDloc-groups (or block-rigid CRQloc-groups) with T = Tcr(X)= Tcr(Y ) and Φ1 : X −→ B and Φ2 : Y −→ C be regular representations of groups B and C with the partition T = Si∈ω Ti and the same coefficients ατ , τ ∈ T . Then B and C are nearly isomorphic. Corollary Near-isomorphism criteria and decomposition theory of (1) B CRQloc-groups in terms of type invariants mτ derive from the theory of CRQloc-groups. 14 15 16 17 18.
Load more

Recommended publications
  • On Abelian Subgroups of Finitely Generated Metabelian
    J. Group Theory 16 (2013), 695–705 DOI 10.1515/jgt-2013-0011 © de Gruyter 2013 On abelian subgroups of finitely generated metabelian groups Vahagn H. Mikaelian and Alexander Y. Olshanskii Communicated by John S. Wilson To Professor Gilbert Baumslag to his 80th birthday Abstract. In this note we introduce the class of H-groups (or Hall groups) related to the class of B-groups defined by P. Hall in the 1950s. Establishing some basic properties of Hall groups we use them to obtain results concerning embeddings of abelian groups. In particular, we give an explicit classification of all abelian groups that can occur as subgroups in finitely generated metabelian groups. Hall groups allow us to give a negative answer to G. Baumslag’s conjecture of 1990 on the cardinality of the set of isomorphism classes for abelian subgroups in finitely generated metabelian groups. 1 Introduction The subject of our note goes back to the paper of P. Hall [7], which established the properties of abelian normal subgroups in finitely generated metabelian and abelian-by-polycyclic groups. Let B be the class of all abelian groups B, where B is an abelian normal subgroup of some finitely generated group G with polycyclic quotient G=B. It is proved in [7, Lemmas 8 and 5.2] that B H, where the class H of countable abelian groups can be defined as follows (in the present paper, we will call the groups from H Hall groups). By definition, H H if 2 (1) H is a (finite or) countable abelian group, (2) H T K; where T is a bounded torsion group (i.e., the orders of all ele- D ˚ ments in T are bounded), K is torsion-free, (3) K has a free abelian subgroup F such that K=F is a torsion group with trivial p-subgroups for all primes except for the members of a finite set .K/.
    [Show full text]
  • Infinite Groups with Large Balls of Torsion Elements and Small Entropy
    INFINITE GROUPS WITH LARGE BALLS OF TORSION ELEMENTS AND SMALL ENTROPY LAURENT BARTHOLDI, YVES DE CORNULIER Abstract. We exhibit infinite, solvable, abelian-by-finite groups, with a fixed number of generators, with arbitrarily large balls consisting of torsion elements. We also provide a sequence of 3-generator non-nilpotent-by-finite polycyclic groups of algebraic entropy tending to zero. All these examples are obtained by taking ap- propriate quotients of finitely presented groups mapping onto the first Grigorchuk group. The Burnside Problem asks whether a finitely generated group all of whose ele- ments have finite order must be finite. We are interested in the following related question: fix n sufficiently large; given a group Γ, with a finite symmetric generating subset S such that every element in the n-ball is torsion, is Γ finite? Since the Burn- side problem has a negative answer, a fortiori the answer to our question is negative in general. However, it is natural to ask for it in some classes of finitely generated groups for which the Burnside Problem has a positive answer, such as linear groups or solvable groups. This motivates the following proposition, which in particularly answers a question of Breuillard to the authors. Proposition 1. For every n, there exists a group G, generated by a 3-element subset S consisting of elements of order 2, in which the n-ball consists of torsion elements, and satisfying one of the additional assumptions: (1) G is solvable, virtually abelian, and infinite (more precisely, it has a free abelian normal subgroup of finite 2-power index); in particular it is linear.
    [Show full text]
  • 17 Torsion Subgroup Tg
    17 Torsion subgroup tG All groups in this chapter will be additive. Thus extensions of A by C can be written as short exact sequences: f g 0 ! A ¡! B ¡! C ! 0 which are sequences of homomorphisms between additive groups so that im f = ker g, ker f = 0 (f is a monomorphism) and coker g = 0 (g is an epimorphism). Definition 17.1. The torsion subgroup tG of an additive group G is defined to be the group of all elements of G of finite order. [Note that if x; y 2 G have order n; m then nm(x + y) = nmx + nmy = 0 so tG · G.] G is called a torsion group if tG = G. It is called torsion-free if tG = 0. Lemma 17.2. G=tG is torsion-free for any additive group G. Proof. This is obvious. If this were not true there would be an element of G=tG of finite order, say x + tG with order n. Then nx + tG = tG implies that nx 2 tG so there is an m > 0 so that m(nx) = 0. But then x has finite order and x + tG is zero in G=tG. Lemma 17.3. Any homomorphism f : G ! H sends tG into tH, i.e., t is a functor. Proof. If x 2 tG then nx = 0 for some n > 0 so nf(x) = f(nx) = 0 which implies that f(x) 2 tH. Lemma 17.4. The functor t commutes with direct sum, i.e., M M t G® = tG®: This is obvious.
    [Show full text]
  • Torsion in Profinite Completions
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 88 (1993) 143-154 143 North-Holland Torsion in profinite completions Peter H. Kropholler School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road. London El 4NS, United Kingdom John S. Wilson Christ’s College, Cambridge CB2 3BCJ, United Kingdom Communicated by A. Camina Received 15 December 1992 For Karl Gruenberg on his 65th birthday 1. Introduction In [l] the question was raised whether the profinite completion of a torsion-free residually finite group is necessarily torsion-free. Subsequently, counter-examples were discovered by Evans [2], and more recently, Lubotsky [4] has shown that there is a finitely generated torsion-free residually finite group whose profinite completion contains a copy of every finite group. In this paper we study the situation more closely for soluble groups. The major result of the paper is the construction of finitely generated centre by metabelian counter-examples in Section 3. We begin in Section 2 by establishing some positive results showing that there are no counter-examples amongst abelian groups, finitely generated abelian by nilpotent groups or soluble minimax groups. In Section 3 we first establish counter-examples which are nilpotent of class 2: these are necessarily infinitely generated. Then we give the promised centre by metabelian examples. It is interesting to note that the dichotomy arising here between abelian by nilpotent groups and centre by metabelian groups is the one which arises in the work of Hall [3].
    [Show full text]
  • Group Theory
    Algebra Math Notes • Study Guide Group Theory Table of Contents Groups..................................................................................................................................................................... 3 Binary Operations ............................................................................................................................................................. 3 Groups .............................................................................................................................................................................. 3 Examples of Groups ......................................................................................................................................................... 4 Cyclic Groups ................................................................................................................................................................... 5 Homomorphisms and Normal Subgroups ......................................................................................................................... 5 Cosets and Quotient Groups ............................................................................................................................................ 6 Isomorphism Theorems .................................................................................................................................................... 7 Product Groups ...............................................................................................................................................................
    [Show full text]
  • The CI Problem for Infinite Groups
    The CI problem for infinite groups Joy Morris∗ Department of Mathematics and Computer Science University of Lethbridge Lethbridge, AB. T1K 3M4. Canada [email protected] Submitted: Feb 24, 2015; Accepted: Nov 21, 2016; Published: Dec 9, 2016 Mathematics Subject Classifications: 05C25, 20B27 Abstract A finite group G is a DCI-group if, whenever S and S0 are subsets of G with the Cayley graphs Cay(G; S) and Cay(G; S0) isomorphic, there exists an automorphism ' of G with '(S) = S0. It is a CI-group if this condition holds under the restricted assumption that S = S−1. We extend these definitions to infinite groups, and make two closely-related definitions: an infinite group is a strongly (D)CIf -group if the same condition holds under the restricted assumption that S is finite; and an infinite group is a (D)CIf -group if the same condition holds whenever S is both finite and generates G. We prove that an infinite (D)CI-group must be a torsion group that is not locally- finite. We find infinite families of groups that are (D)CIf -groups but not strongly (D)CIf -groups, and that are strongly (D)CIf -groups but not (D)CI-groups. We discuss which of these properties are inherited by subgroups. Finally, we completely n characterise the locally-finite DCI-graphs on Z . We suggest several open problems related to these ideas, including the question of whether or not any infinite (D)CI- group exists. 1 Introduction Although there has been considerable work done on the Cayley Isomorphism problem for finite groups and graphs, little attention has been paid to its extension to the infinite case.
    [Show full text]
  • High Subgroups of Abelian Torsion Groups
    Pacific Journal of Mathematics HIGH SUBGROUPS OF ABELIAN TORSION GROUPS JOHN MCCORMICK IRWIN Vol. 11, No. 4 1961 HIGH SUBGROUPS OF ABELIAN TORSION GROUPS JOHN M. IRWIN The results in this paper were part of a doctor's thesis completed in February 1960 under Professor W. R. Scott at the University of Kansas. The author wishes to express his gratitude to Professor Scott for his advice and for checking the results. In what follows, all groups considered are Abelian. Let G1 be the subgroup of elements of infinite height in an Abelian group G (see [2]). A subgroup H of G maximal with respect to disjointness from G1 will be called a high subgroup of G. If N is a subgroup of G, H will be called N-high if and only if H is a subgroup of G maximal with respect to disjointness from N. Zorn's lemma guarantees the existence of N- high subgroups for any subgroup N of G. A group E minimal divisible among those groups containing G will be called a divisible hull of G. Unless otherwise specified, the notation and terminology will be essentially that of L. Fuchs in [1]. The main theorem says that high subgroups of Abelian torsion groups are pure. After proving some preparatory lemmas, we will prove the main theorem. Then we will discuss Fuchs' Problem 4 and list some of the more important properties of high subgroups. Finally we will state some generalizations. A lemma describing iV-high subgroups is LEMMA 1. Let G be a primary group with H an N-high subgroup of G, Da divisible hull of G, A any divisible hull of H in D (this means that A c D), and B any divisible hull of N in D.
    [Show full text]
  • The Torsion Group of a Field Defined by Radicals
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF NUMBER THEORY 19, 283-294 (1984) The Torsion Group of a Field Defined by Radicals MARIA ACOSTA DE OROZCO AND WILLIAM YSLAS VELEZ* Department of Mathematics, University of Arizona, Tucson, Arizona 85721 Communicated by 0. Taussky Todd Received February 24, 1983 Let L/F be a finite separable extension, L* =L\(O}. and T&*/F*) the torsion subgroup of L*/F*. When L/F is an abelian extension T(L*/F*) is explicitly determined. This information is used to study the structure of T(L*/F*). In particular, T(F(a)*/F*) when om = a E F is explicitly determined. c 1984 Academic Press. Inc. 1. INTRODUCTION Let L/F be a finite separable extension and let L* denote the multiplicative group of non-zero elements. We shall denote by T = T(L */F*) the torsion subgroup of L*/F*, that is, T= {/?F* : ,L?E L and /3k E F for some integer k}. The group T has been studied for various extensions L/F. Kummer theory yields that if L/F is normal, Gal(L/F) is abelian of exponent m (that is, every element in Gal(L/F) has order a divisor of m) and F contains m distinct mth roots of unity, then {aF*: CI E L and am E F) is isomorphic to Gal(L/F). More generally, Kneser (51 has proved the following: THEOREM (Kneser). Let L/F be a jkite separable extension, N a subgroup of L* containing F* and IN/F* 1< CD.
    [Show full text]
  • On Torsion and Finite Extension of Fc and Τnk Groups in Certain Classes of Finitely Generated Groups
    Open J. Math. Sci., Vol. 2(2018), No. 1, pp. 351 - 360 Website: https://pisrt.org/psr-press/journals/oms/ ISSN: 2523-0212 (Online) 2616-4906 (Print) http://dx.doi.org/10.30538/oms2018.0040 ON TORSION AND FINITE EXTENSION OF FC AND τNK GROUPS IN CERTAIN CLASSES OF FINITELY GENERATED GROUPS MOURAD CHELGHAM, MOHAMED KERADA1 (2) Abstract. Let k > 0 an integer. F , τ, N, Nk, Nk and A denote the classes of finite, torsion, nilpotent, nilpotent of class at most k, group which every two generator subgroup is Nk and abelian groups respectively. The main results of this paper is, firstly, we prove that, in the class of finitely generated τN-group (respectively FN-group) a (FC)τ-group (respectively (FC)F -group) is a τA-group (respectively is FA-group). Secondly, we prove that a finitely generated τN-group (respectively FN-group) in the (2) class ((τNk)τ; 1) (respectively ((FNk)F; 1)) is a τNk -group (respec- (2) tively FNk -group). Thirdly we prove that a finitely generated τN-group ∗ (respectively FN-group) in the class ((τNk)τ; 1) ( respectively ((FNk)F; ∗ 1) ) is a τNc-group (respectively FNc-group) for certain integer c and we extend this results to the class of NF -groups. Mathematics Subject Classification: 20F22, 20F99. Key words and phrases: (FC)τ-group; (FC)F -group; ((τNk)τ; 1)-group; ∗ ∗ ((FNk)F; 1)-group; ((τNk)τ; 1) -group; ((FNk)F; 1) -group. 1. Introduction and Preliminaries Definition 1.1. A group G is said to be with finite conjugacy classes (or shortly FC-group) if and only if every element of G has a finite conjugacy class in G.
    [Show full text]
  • Selected Exercises from Abstract Algebra by Dummit and Foote (3Rd Edition)
    Selected exercises from Abstract Algebra by Dummit and Foote (3rd edition). Bryan F´elix Abril 12, 2017 Section 2.1 Exercise (6). Let G be an abelian group. Prove that T = fg 2 Gjjgj < 1g is a subgroup of G. Give an explicit example where this set is not a subgroup when G is non-abelian. Proof. i. T is non-empty. The identity i of G is an element of T . ii. T is closed under inverses. If x 2 T then there exist n < 1 such that xn = i. Then i = xn x−1 = xnx−1 (x−1)n = (xnx−1)n = ix−n since x commutes with itself = xnx−n since xn = i = i This shows that jx−1j ≤ n and therefore x−1 2 T . iii. T is closed under products. Take arbitrary x and y in T and let jxj = n and jyj = m. Then (xy)mn = xmnymn since G is an abelian group = (xn)m(ym)n = imin = i Therefore jxyj ≤ mn and xy 2 T We conclude thhat T is a subgroup. Now we exhibit an example where T fails to be a subgroup when G is non-abelian. Let G = h x; yjx2 = y2 = i i. Observe that the product xy has infinite order since the product (xy)(xy) is no longer reducible because we lack the commutativity of x with y. Hence, x; y 2 T but (xy) 2= T . Exercise (7). Fix some n 2 Z with n > 1. Find the torsion subgroup of Z×(Z=nZ). Show that the set of elements of infinite order together with the identity is not a subgroup of this direct product.
    [Show full text]
  • Arxiv:1802.01098V1 [Math.AG] 4 Feb 2018 Iet Ihm Ece Ute Raiescesi Ler Ntenex [ the Plotkin in B
    GEOMETRIC EQUIVALENCE of π-TORSION-FREE NILPOTENT GROUPS. R. Lipyanski This article is dedicated to my teacher Prof. B. Plotkin on his 90th anniversary. Abstract. In this paper we study the property of geometric equivalence of groups introduced by B. Plotkin [P1, P2]. Sufficient and necessary conditions are presented for π-torsion-free nilpotent group to be geometrically equiva- lent to its π-completion. We prove that a relatively free nilpotent π-torsion- free group and its π-completion define the same quasi-variety. Examples of π-torsion-free nilpotent groups that are geometrically equivalent to their π- completions are given. 1. Introduction I attended Professor Plotkin’s lectures about 45 years ago as a student at the University of Latvia. Over the years, I participated in Plotkin’s seminars on group theory in Latvia and Israel. I am grateful to Boris Isaakovich Plotkin for revealing to me the world of group theory in all its diversity. The problems posed by Plotkin at seminars and in his numerous papers have always been interesting to me. I would like to wish my teacher further creative success in Algebra in the next 30 years. The foundation of universal algebraic geometry for different algebraic systems were laid by B. Plotkin [P1, P2, P3]. Within the framework of universal algebraic geometry (UAG), the important notion of geometric equivalence of universal alge- arXiv:1802.01098v1 [math.AG] 4 Feb 2018 bras was introduced in [P1, P2]. Geometric equivalence of two algebras means, in some sense, equal possibilities for solving systems of equations in these algebras.
    [Show full text]
  • Centralizers in Residually Finite Torsion Groups
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 12, December 1998, Pages 3495{3499 S 0002-9939(98)04471-2 CENTRALIZERS IN RESIDUALLY FINITE TORSION GROUPS ANER SHALEV (Communicated by Lance W. Small) In memory of Brian Hartley Abstract. Let G be a residually finite torsion group. We show that, if G has a finite 2-subgroup whose centralizer is finite, then G is locally finite. We also show that, if G has no 2-torsion, and Q is a finite 2-group acting on G in such a way that the centralizer CG(Q) is soluble, or of finite exponent, then G is locally finite. 1. Introduction The study of centralizers in torsion groups and in locally finite groups in partic- ular has been a major research project in the past few decades; see Shunkov [S], Hartley [H1], [H2], [H3, Section 3] and the references therein. The purpose of this note is to apply some powerful Lie-theoretic tools (see Zelmanov [Z2] and Bahturin and Zaicev [BZ]) in the study of centralizers in residually finite torsion groups. Well known constructions by Golod [G], Grigorchuk [Gr], Gupta and Sidki [GS], and others, show that a residually finite torsion group G need not be locally finite. However, the local finiteness of G can be deduced under some extra assumptions on G. For example, if we assume further that G is soluble, or of finite exponent, then G is locally finite; note that the fact residually finite groups of finite exponent are locally finite is one of the equivalent formulations of Zelmanov's celebrated solution to the Restricted Burnside Problem.
    [Show full text]