Countable Torsion Products of Abelian /^-Groups E
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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/. -
Nearly Isomorphic Torsion Free Abelian Groups
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 35, 235-238 (1975) Nearly Isomorphic Torsion Free Abelian Groups E. L. LADY University of Kansas, Lawrence, Kansas 66044* Communicated by D. Buchsbaum Received December 7, 1973 Let K be the Krull-Schmidt-Grothendieck group for the category of finite rank torsion free abelian groups. The torsion subgroup T of K is determined and it is proved that K/T is free. The investigation of T leads to the concept of near isomorphism, a new equivalence relation for finite rank torsion free abelian groups which is stronger than quasiisomorphism. If & is an additive category, then the Krull-Schmidt-Grothendieck group K(d) is defined by generators [A], , where A E &, and relations [A]& = m-2 + [af? Pwhenever A w B @ C. It is well-known that every element in K(&‘) can be written in the form [A],cJ - [Bls/ , and that [A],, = [B], ifandonlyifA @L = B @LforsomeLE.&. We let ,F denote the category of finite rank torsion free abelian groups, and we write K = K(3). We will write [G] rather than [G],7 for the class of [G] in K. If G, H ~9, then Hom(G, H) and End(G) will have the usual significance. For any other category ~9’, &(G, H) and d(G) will denote the corresponding group of $Z-homomorphisms and ring of cpJ-endomorphisms. We now proceed to define categories M and NP, where p is a prime number or 0. -
THE TORSION in SYMMETRIC POWERS on CONGRUENCE SUBGROUPS of BIANCHI GROUPS Jonathan Pfaff, Jean Raimbault
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Archive Ouverte en Sciences de l'Information et de la Communication THE TORSION IN SYMMETRIC POWERS ON CONGRUENCE SUBGROUPS OF BIANCHI GROUPS Jonathan Pfaff, Jean Raimbault To cite this version: Jonathan Pfaff, Jean Raimbault. THE TORSION IN SYMMETRIC POWERS ON CONGRUENCE SUBGROUPS OF BIANCHI GROUPS. Transactions of the American Mathematical Society, Ameri- can Mathematical Society, 2020, 373 (1), pp.109-148. 10.1090/tran/7875. hal-02358132 HAL Id: hal-02358132 https://hal.archives-ouvertes.fr/hal-02358132 Submitted on 11 Nov 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THE TORSION IN SYMMETRIC POWERS ON CONGRUENCE SUBGROUPS OF BIANCHI GROUPS JONATHAN PFAFF AND JEAN RAIMBAULT Abstract. In this paper we prove that for a fixed neat principal congruence subgroup of a Bianchi group the order of the torsion part of its second cohomology group with coefficients in an integral lattice associated to the m-th symmetric power of the standard 2 representation of SL2(C) grows exponentially in m . We give upper and lower bounds for the growth rate. -
BLECHER • All Numbers Below Are Integers, Usually Positive. • Read Up
MATH 4389{ALGEBRA `CHEATSHEET'{BLECHER • All numbers below are integers, usually positive. • Read up on e.g. wikipedia on the Euclidean algorithm. • Diophantine equation ax + by = c has solutions iff gcd(a; b) doesn't divide c. One may find the solutions using the Euclidean algorithm (google this). • Euclid's lemma: prime pjab implies pja or pjb. • gcd(m; n) · lcm(m; n) = mn. • a ≡ b (mod m) means mj(a − b). An equivalence relation (reflexive, symmetric, transitive) on the integers. • ax ≡ b (mod m) has a solution iff gcd(m; a)jb. • Division Algorithm: there exist unique integers q and r with a = bq + r and 0 ≤ r < b; so a ≡ r (mod b). • Fermat's theorem: If p is prime then ap ≡ a (mod p) Also, ap−1 ≡ 1 (mod p) if p does not divide a. • For definitions for groups, rings, fields, subgroups, order of element, etc see Algebra fact sheet" below. • Zm = f0; 1; ··· ; m−1g with addition and multiplication mod m, is a group and commutative ring (= Z =m Z) • Order of n in Zm (i.e. smallest k s.t. kn ≡ 0 (mod m)) is m=gcd(m; n). • hai denotes the subgroup generated by a (the smallest subgroup containing a). For example h2i = f0; 2; 2+2 = 4; 2 + 2 + 2 = 6g in Z8. (In ring theory it may be the subring generated by a). • A cyclic group is a group that can be generated by a single element (the group generator). They are abelian. • A finite group is cyclic iff G = fg; g + g; ··· ; g + g + ··· + g = 0g; so G = hgi. -
Math 521 – Homework 5 Due Thursday, September 26, 2019 at 10:15Am
Math 521 { Homework 5 Due Thursday, September 26, 2019 at 10:15am Problem 1 (DF 5.1.12). Let I be any nonempty index set and let Gi be a group for each i 2 I. The restricted direct product or direct sum of the groups Gi is the set of elements of the direct product which are the identity in all but finitely many components, that is, the Q set of elements (ai)i2I 2 i2I Gi such that ai = 1i for all but a finite number of i 2 I, where 1i is the identity of Gi. (a) Prove that the restricted direct product is a subgroup of the direct product. (b) Prove that the restricted direct produce is normal in the direct product. (c) Let I = Z+, let pi denote the ith prime integer, and let Gi = Z=piZ for all i 2 Z+. Show that every element of the restricted direct product of the Gi's has finite order but the direct product Q G has elements of infinite order. Show that in this i2Z+ i example, the restricted direct product is the torsion subgroup of Q G . i2Z+ i Problem 2 (≈DF 5.5.8). (a) Show that (up to isomorphism), there are exactly two abelian groups of order 75. (b) Show that the automorphism group of Z=5Z×Z=5Z is isomorphic to GL2(F5), where F5 is the field Z=5Z. What is the size of this group? (c) Show that there exists a non-abelian group of order 75. (d) Show that there is no non-abelian group of order 75 with an element of order 25. -
On the Growth of Torsion in the Cohomology of Arithmetic Groups
ON THE GROWTH OF TORSION IN THE COHOMOLOGY OF ARITHMETIC GROUPS A. ASH, P. E. GUNNELLS, M. MCCONNELL, AND D. YASAKI Abstract. Let G be a semisimple Lie group with associated symmetric space D, and let Γ G be a cocompact arithmetic group. Let L be a lattice in- side a ZΓ-module⊂ arising from a rational finite-dimensional complex representa- tion of G. Bergeron and Venkatesh recently gave a precise conjecture about the growth of the order of the torsion subgroup Hi(Γk; L )tors as Γk ranges over a tower of congruence subgroups of Γ. In particular they conjectured that the ra- tio log Hi(Γk; L )tors /[Γ:Γk] should tend to a nonzero limit if and only if i = (dim(D|) 1)/2 and G| is a group of deficiency 1. Furthermore, they gave a precise expression− for the limit. In this paper, we investigate computationally the cohomol- ogy of several (non-cocompact) arithmetic groups, including GLn(Z) for n = 3, 4, 5 and GL2(O) for various rings of integers, and observe its growth as a function of level. In all cases where our dataset is sufficiently large, we observe excellent agree- ment with the same limit as in the predictions of Bergeron–Venkatesh. Our data also prompts us to make two new conjectures on the growth of torsion not covered by the Bergeron–Venkatesh conjecture. 1. Introduction 1.1. Let G be a connected semisimple Q-group with group of real points G = G(R). Suppose Γ G(Q) is an arithmetic subgroup and M is a ZΓ-module arising from a rational finite-dimensional⊂ complex representation of G. -
Countable Torsion-Free Abelian Groups
COUNTABLE TORSION-FREE ABELIAN GROUPS By M. O'N. CAMPBELL [Received 27 January 1959.—Read 19 February 1959] Introduction IN this paper we present a new classification of the countable torsion-free abelian groups. Since any abelian group 0 may be regarded as an extension of a periodic group (namely the torsion part of G) by a torsion-free group, and since there exists the well-known complete classification of the count- able periodic abelian groups due to Ulm, it is clear that the classification problem studied here is of great interest. By a group we shall always mean an abelian group, and the additive notation will be employed throughout. It is well known that all the maximal linearly independent sets of elements of a given group are car- dinally equivalent; the corresponding cardinal number is the rank of the group. Derry (2) and Mal'cev (4) have studied the torsion-free groups of finite rank only, and have given classifications of these groups in terms of certain equivalence classes of systems of matrices with £>-adic elements. Szekeres (5) attempted to abolish the finiteness restriction on rank; he extracted certain invariants, but did not derive a complete classification. By a basis of a torsion-free group 0 we mean any maximal linearly independent subset of G. A subgroup generated by a basis of G will be called a basal subgroup or described as basal in G. Thus U is a basal sub- group of G if and only if (i) U is free abelian, and (ii) the factor group G/U is periodic. -
Arxiv:1909.05807V1 [Math.RA] 12 Sep 2019 Cs T
MODULES OVER TRUSSES VS MODULES OVER RINGS: DIRECT SUMS AND FREE MODULES TOMASZ BRZEZINSKI´ AND BERNARD RYBOLOWICZ Abstract. Categorical constructions on heaps and modules over trusses are con- sidered and contrasted with the corresponding constructions on groups and rings. These include explicit description of free heaps and free Abelian heaps, coprod- ucts or direct sums of Abelian heaps and modules over trusses, and description and analysis of free modules over trusses. It is shown that the direct sum of two non-empty Abelian heaps is always infinite and isomorphic to the heap associated to the direct sums of the group retracts of both heaps and Z. Direct sum is used to extend a given truss to a ring-type truss or a unital truss (or both). Free mod- ules are constructed as direct sums of a truss. It is shown that only free rank-one modules are free as modules over the associated truss. On the other hand, if a (finitely generated) module over a truss associated to a ring is free, then so is the corresponding quotient-by-absorbers module over this ring. 1. Introduction Trusses and skew trusses were defined in [3] in order to capture the nature of the distinctive distributive law that characterises braces and skew braces [12], [6], [9]. A (one-sided) truss is a set with a ternary operation which makes it into a heap or herd (see [10], [11], [1] or [13]) together with an associative binary operation that distributes (on one side or both) over the heap ternary operation. If the specific bi- nary operation admits it, a choice of a particular element could fix a group structure on a heap in a way that turns the truss into a ring or a brace-like system (which becomes a brace provided the binary operation admits inverses). -
7 Torsion Subgroups and Endomorphism Rings
18.783 Elliptic Curves Spring 2019 Lecture #7 02/27/2019 7 Torsion subgroups and endomorphism rings 7.1 The n-torsion subgroup E[n] Having determined the degree and separability of the multiplication-by-n map [n] in the previous lecture, we now want to determine the structure of its kernel, the n-torsion subgroup E[n], as a finite abelian group. Recall that any finite abelian group G can be written as a direct sum of cyclic groups of prime power order (unique up to ordering). Since #E[n] always divides deg[n] = n2, to determine the structure of E[n] it suffices to determine the structure of E[`e] for each prime power `e dividing n. Theorem 7.1. Let E=k be an elliptic curve and let p := char(k). For each prime `: ( =`e ⊕ =`e if ` 6= p; E[`e] ' Z Z Z Z e Z=` Z or f0g if ` = p: Proof. We first suppose ` 6= p. The multiplication-by-` map [`] is then separable, and we may apply Theorem 6.8 to compute #E[`] = # ker[`] = deg[`] = `2. Every nonzero element e of E[`] has order `, so we must have E[`] ' Z=`Z ⊕ Z=`Z. If E[` ] ' hP1i ⊕ · · · ⊕ hPri with e each Pi 2 E(k¯) of order ` i > 1, then e −1 e −1 r E[`] ' h` 1 P i ⊕ · · · ⊕ h` r P i ' (Z=`Z) ; and we must have r = 2; more generally, for any abelian group G the `-rank r of G[`e] e e e is the same as the `-rank of G[`]. -
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. -
Torsion Subgroups of Elliptic Curves Over Quadratic Cyclotomic Fields in Elementary Abelian 2-Extensions
TORSION SUBGROUPS OF ELLIPTIC CURVES OVER QUADRATIC CYCLOTOMIC FIELDS IN ELEMENTARY ABELIAN 2-EXTENSIONS OZLEM¨ EJDER p Abstract. Let K denote the quadratic field Q( d) where d = −1 or −3 and let E be an elliptic curve defined over K. In this paper, we analyze the torsion subgroups of E in the maximal elementary abelian 2-extension of K. 1. Introduction Finding the set of rational points on a curve is one of the fundamental problems in number theory. Given a number field K and an algebraic curve C=K, the set, C(K), of points on C which are defined over K, has the following properties depending on the genus of the curve. (1) If C has genus 0, then C(Q) is either empty or infinite. (2) If C has genus greater than 1, then C(Q) is either empty or finite. (Faltings's theorem) Assume C has genus 1. If C(K) is not empty, then it forms a finitely generated abelian group, proven by Luis Mordell and Andr´eWeil. Thus, given an elliptic curve E=K, the group E(K) has the structure r E(K)tors ⊕ Z : Here E(K)tors is the finite part of this group E(K), called the torsion sub- group and the integer r is called the rank of E over the number field K. In this paper, we will study the elliptic curves over the quadratic cyclo- tomic fields and how their torsion subgroups grow in the compositum of all quadratic extensions of the base field. First, we summarize the results obtained so far on E(K)tors for a number field K. -
The Torsion Index of the Spin Groups
The torsion index of the spin groups Burt Totaro The torsion index is a positive integer associated by Grothendieck to any con- nected compact Lie group G [10]. As explained in section 1, knowing the torsion index of a group has direct consequences for the abelian subgroups of G, the inte- gral cohomology of the classifying space, the complex cobordism of the classifying space, the Chow ring of the classifying space [26], and the classification of G-torsors over fields [24]. Demazure [5], Marlin [14], and Tits [25] have given upper bounds for the torsion index. In this paper, we compute the torsion index exactly for all the spin groups Spin(n). In another paper, we will compute the torsion index of the exceptional group E8. As discussed below, this completes the calculation of the torsion index for all the simply connected simple groups. The topology of the spin groups becomes more and more complicated as the dimension increases, and so it was not at all clear that it would be possible to do the calculation in all dimensions. Indeed, the answer is rather intricate, and the proof in high dimensions requires some deep information from√ analytic number theory, Bauer and Bennett’s theorem on the binary expansion of 2 [1]. Theorem 0.1 Let l be a nonnegative integer. The groups Spin(2l+1) and Spin(2l+ 2) have the same torsion index, of the form 2u(l). For all l, u(l) is either l + 1 l − log + 1 2 2 or that expression plus 1. The second case arises only for certain numbers l (ini- tially: l = 8, 16, 32, 33,...) which are equal to or slightly larger than a power of 2.