DOCSLIB.ORG

On Optimal Scott Sentences of Finitely Generated Algebraic Structures

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On Optimal Scott Sentences of Finitely Generated Algebraic Structures PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000{000 S 0002-9939(XX)0000-0 ON OPTIMAL SCOTT SENTENCES OF FINITELY GENERATED ALGEBRAIC STRUCTURES MATTHEW HARRISON-TRAINOR AND MENG-CHE HO (Communicated by Heike Mildenberger) Abstract. Scott showed that for every countable structure A, there is a sen- tence of the infinitary logic L!1!, called a Scott sentence for A, whose count- able models are exactly the isomorphic copies of A. Thus, the least quantifier complexity of a Scott sentence of a structure is an invariant that measures the complexity \describing" the structure. Knight et al. have studied the Scott sentences of many structures. In particular, Knight and Saraph showed that a 0 finitely generated structure always has a Σ3 Scott sentence. We give a charac- 0 terization of the finitely generated structures for which the Σ3 Scott sentence is optimal. One application of this result is to give a construction of a finitely 0 generated group where the Σ3 Scott sentence is optimal. 1. Introduction Given a countable structure M, we can describe M, up to isomorphism, by a sentence of the infinitary logic L!1! which allows countable conjunctions and disjunctions. (See Section 1.1 for the formal description of this logic; in this brief introduction, we will write down sentences in an informal way.) To measure the complexity of M, we want to write down the simplest possible description of M. For example, one can describe the countably infinite-dimensional Q-vector space by the vector space axioms together with the sentence for all n, there are x1; : : : ; xn such that for all r1; : : : ; rn 2 Q, if r1x1+···+rnxn = 0 then some ri = 0. This sentence has a universal quantifier, followed by an existential quantifier, fol- lowed by a universal quantifier. There is a hierarchy of sentences depending on the 0 number of quantifier alternations. The Σn sentences have n alternations of quan- 0 tifiers, beginning with existential quantifiers; the Πn sentences have n alternations 0 of quantifiers, beginning with universal quantifiers; and the d-Σn sentences are the 0 0 conjunction of a Σn and a Πn sentence. The hierarchy is ordered as follows, from Received by the editors December 29, 2017. 2010 Mathematics Subject Classification. Primary 03D45, 03C57, 20F10; Secondary 20F06, 20E06. c XXXX American Mathematical Society 1 2 MATTHEW HARRISON-TRAINOR AND MENG-CHE HO the simplest formulas on the left, to the most complicated formulas on the right: Σ0 Σ0 Σ0 9 1 9 2 9 3 0 0 " 0 0 0 " 0 0 0 " 0 Σ1 \ Π1 d-Σ1 / Σ2 \ Π2 d-Σ2 / Σ3 \ Π3 d-Σ3 / ··· < < < % 0 % 0 % 0 Π1 Π2 Π3 We use this hierarchy to measure the complexity of a sentence. The sentence given 0 above describing the infinite-dimensional Q-vector space is a Π3 sentence, and it 0 turns out that this is the best possible; there is no d-Σ2 description of this vector 0 space. There is a d-Σ2 description of any finite-dimensional Q-vector space, and so these structures are \simpler" than the infinite-dimensional vectors space. In this paper, we consider descriptions of finitely generated structures, and par- ticularly of finitely generated groups. Any finitely generated structure M, with 0 generating tuplea ¯, has a Σ3 description of the form: there is a tuple x¯, satisfying the same atomic formulas as a¯ (i.e., for all atomic formulas and negations of atomic formulas true of a¯, the formula is true of x¯), such that every element is generated by x¯ (i.e., for all y, there is a term t in the language such that y = t(¯x)). 0 However, many finitely generated groups have a simpler description which is d-Σ2. 0 For the group Z, for example, the Π2 axioms of torsion-free abelian groups, together 0 0 0 with the following two sentences, which are Π2 and Σ2 respectively, form a d-Σ2 description: for all x and y, there are n; m 2 Z, not both zero, such that nx = my and there is x 6= 0 which has no proper divisors. 0 Indeed, all previously known examples of finitely generated groups had a d-Σ2 Scott sentences, including all polycyclic (including nilpotent) groups and many finitely generated solvable groups [5]. The main result of this paper is an example of a 0 computable finitely generated group that has no d-Σ2 Scott sentence. Our group 0 has Σ3 m-complete index set. This paper is divided into two main sets of results. The first is a general investi- 0 gation of conditions for a finitely generated structure to have (or not have) a d-Σ2 Scott sentence. The second is an application of these general results to construct- ing the group mentioned above. We also include some results on finitely generated fields and rings. 1.1. Scott sentences. The infinitary logic L!1! is the logic which allows count- ably infinite conjunctions and disjunctions but only finite quantification. If the conjunctions and disjunctions of a formula ' are all over computable sets of indices for formulas, then we say that ' is computable. We use the following recursive definition to define the complexity of classes: 0 0 • An L!1! formula is both Σ0 and Π0 if it is quantifier free and does not contain any infinite disjunction or conjunction. 0 • An L!1! formula is Σα if it is a countable disjunction of formulas of the 0 form 9xφ¯ where each φ is Πβ for some β < α. 0 • An L!1! formula is Πα if it is a countable disjunction of formulas of the 0 form 8xφ¯ where each φ is Σβ for some β < α. SCOTT SENTENCES OF F.G. STRUCTURES 3 0 0 0 We say a formula is d-Σα if it is a conjunction of a Σα formula and a Πα formula. Scott [10] showed that if A is a countable structure in a countable language, then there is a sentence ' of L!1! whose countable models are exactly the isomorphic copies of A. Such a sentence is called a Scott sentence for A. We remark that because α ^ :(β ^ :γ) is equivalent to (α ^ :β) _ (α ^ γ), the complexity classes 0 n-Σα of Scott sentences collapse for n ≥ 2. We can measure the complexity of a countable structure by looking for a Scott sentence of minimal complexity, as measured by the quantifier complexity hierarchy 0 of computable formulas described above. [9] showed that if A has a Πα Scott 0 0 sentence and a Σα Scott sentence, then it must have a d-Σβ Scott sentence for 0 0 some β < α. So for a given structure, the optimal Scott sentence is Σα,Πα, or 0 d-Σα for some α. We refer the interested readers to Chapter 6 of [2] for a more complete description of L!1! formulas and Scott sentences. 1.2. Index set complexity. Given a structure A and a Scott sentence ' for A, we want to determine whether ' is an optimal Scott sentence for A, or whether there is a simpler Scott sentence which we have not yet found. We can use index set calculations to resolve this problem. Definition 1.1. Let A be a structure. The index set I(A) is the set of all indices e such that the eth Turing machine Φe gives the atomic diagram of a structure B isomorphic to A. We can also relativize this to any set X: IX (A) is the set of X all indices e such that the eth Turing machine Φe with oracle X gives the atomic diagram of a structure B isomorphic to A. There is a connection between index sets and Scott sentences: 0 Proposition 1.2. If a countable structure A has an X-computable Σα (respectively 0 0 X 0 Πα or d-Σα) Scott sentence, then the index set I (A) is in Σα(X) (respectively 0 0 Πα(X) or d-Σα(X)). 0 So if, for example, we have a computable Σ3 Scott sentence for a structure A, we 0 will try to show that the index set I(A) is Σ3 m-complete. If we can do this, then we know that our Scott sentence is optimal. In general, any L!1! sentence is X-computable for some X. 1.3. Summary of prior results. There are many results using the strategy above to find the complexities of optimal Scott sentences of structures. For example, Knight et al. [3], [4] determined the complexities of optimal Scott sentences for finitely generated free abelian groups, reduced abelian groups, free groups, and many other structures. However, this strategy does not work when the complexity of the optimal Scott sentence is strictly higher than the complexity of the index set. Indeed, Knight and McCoy gave the first such example in [6], showing there is a subgroup G of Q such 0 0 that I(G) is d-Σ2, but it has no computable d-Σ2 Scott sentence. It was observed in [7] that any computable finitely generated group, and indeed 0 any computable finitely generated structure, has a computable Σ3 Scott sentence. In [5], it was shown that many classes of \nice" groups in the sense of geomet- ric group theory, including polycyclic groups (which includes nilpotent groups and 4 MATTHEW HARRISON-TRAINOR AND MENG-CHE HO 0 abelian groups), and certain solvable groups all have computable d-Σ2 Scott sen- 0 tences. However, none of these examples achieves the Σ3 bound that was given in [7].
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]
  • Filling Functions Notes for an Advanced Course on the Geometry of the Word Problem for Finitely Generated Groups Centre De Recer
    Filling Functions Notes for an advanced course on The Geometry of the Word Problem for Finitely Generated Groups Centre de Recerca Mathematica` Barcelona T.R.Riley July 2005 Revised February 2006 Contents Notation vi 1Introduction 1 2Fillingfunctions 5 2.1 Van Kampen diagrams . 5 2.2 Filling functions via van Kampen diagrams . .... 6 2.3 Example: combable groups . 10 2.4 Filling functions interpreted algebraically . ......... 15 2.5 Filling functions interpreted computationally . ......... 16 2.6 Filling functions for Riemannian manifolds . ...... 21 2.7 Quasi-isometry invariance . .22 3Relationshipsbetweenfillingfunctions 25 3.1 The Double Exponential Theorem . 26 3.2 Filling length and duality of spanning trees in planar graphs . 31 3.3 Extrinsic diameter versus intrinsic diameter . ........ 35 3.4 Free filling length . 35 4Example:nilpotentgroups 39 4.1 The Dehn and filling length functions . .. 39 4.2 Open questions . 42 5Asymptoticcones 45 5.1 The definition . 45 5.2 Hyperbolic groups . 47 5.3 Groups with simply connected asymptotic cones . ...... 53 5.4 Higher dimensions . 57 Bibliography 68 v Notation f, g :[0, ∞) → [0, ∞)satisfy f ≼ g when there exists C > 0 such that f (n) ≤ Cg(Cn+ C) + Cn+ C for all n,satisfy f ≽ g ≼, ≽, ≃ when g ≼ f ,andsatisfy f ≃ g when f ≼ g and g ≼ f .These relations are extended to functions f : N → N by considering such f to be constant on the intervals [n, n + 1). ab, a−b,[a, b] b−1ab, b−1a−1b, a−1b−1ab Cay1(G, X) the Cayley graph of G with respect to a generating set X Cay2(P) the Cayley 2-complex of a
    [Show full text]
  • Finitely Generated Groups Are Universal
    Finitely Generated Groups Are Universal Matthew Harrison-Trainor Meng-Che Ho December 1, 2017 Abstract Universality has been an important concept in computable structure theory. A class C of structures is universal if, informally, for any structure, of any kind, there is a structure in C with the same computability-theoretic properties as the given structure. Many classes such as graphs, groups, and fields are known to be universal. This paper is about the class of finitely generated groups. Because finitely generated structures are relatively simple, the class of finitely generated groups has no hope of being universal. We show that finitely generated groups are as universal as possible, given that they are finitely generated: for every finitely generated structure, there is a finitely generated group which has the same computability-theoretic properties. The same is not true for finitely generated fields. We apply the results of this investigation to quasi Scott sentences. 1 Introduction Whenever we have a structure with interesting computability-theoretic properties, it is natural to ask whether such examples can be found within particular classes. While one could try to adapt the proof within the new class, it is often simpler to try and code the original structure into a structure in the given class. It has long been known that for certain classes, such as graphs, this is always possible. Hirschfeldt, Khoussainov, Shore, and Slinko [HKSS02] proved that classes of graphs, partial orderings, lattices, integral domains, and 2-step nilpotent groups are \complete with respect to degree spectra of nontrivial structures, effective dimensions, expansion by constants, and degree spectra of relations".
    [Show full text]
  • Decomposing Some Finitely Generated Groups Into Free Products With
    Decomposing some finitely generated groups into free products with amalgamation 1 V.V. Benyash-Krivets Introduction . We shall say that a group G is a non-trivial free product with amalgamation if ¡ ¡ G = G1 A G2, where G1 = A = G2 (see [1]). Wall [2] has posed the following question: Which one-relator groups are non-trivial free products with amalgamation? £ Let G = ¢ g , . , g R = = R = 1 be a group with m generators and n 1 m | 1 · · · n relations such that def G = m n ¤ 2. It is proved in [4] that G is a non-trivial − free product with amalgamation. In particular, if G is a group with m ¤ 3 generators and one relation, then G is a non-trivial free product with amalgamation. A case of groups with two generators and one relation is more complicated. For example, the free −1 −1 £ abelian group G = ¢ a, b [a, b] = 1 of rank 2, where [a, b] = aba b , obviously, is | not a non-trivial free product with amalgamation. Other examples are given by groups −1 n £ Gn = ¢ a, b aba = b . For any n, the group Gn is solvable and using results from [3], | ¡ it is easy to show that for n = 1 the group G is not decomposable into a non-trivial − n free product with amalgamation. The following conjecture was stated in [4]. m £ ¤ Conjecture 1 Let G = ¢ a, b R (a, b) = 1 , m 2, be a group with two generators | and one relation with torsion. Then G is a non-trivial free product with amalgamation.
    [Show full text]
  • Lecture 4, Geometric and Asymptotic Group Theory
    Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Quasi-isometry Limit groups Free actions Lecture 4, Geometric and asymptotic group theory Olga Kharlampovich NYC, Sep 16 1 / 32 Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Homomorphisms of groups Quasi-isometry Limit groups Free actions The universal property of free groups allows one to describe arbitrary groups in terms of generators and relators. Let G be a group with a generating set S. By the universal property of free groups there exists a homomorphism ': F (S) ! G such that '(s) = s for s 2 S. It follows that ' is onto, so by the first isomorphism theorem G ' F (S)=ker('): In this event ker(') is viewed as the set of relators of G, and a group word w 2 ker(') is called a relator of G in generators S. If a subset R ⊂ ker(') generates ker(') as a normal subgroup of F (S) then it is termed a set of defining relations of G relative to S. 2 / 32 Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Homomorphisms of groups Quasi-isometry Limit groups Free actions The pair hS j Ri is called a presentation of G, it determines G uniquely up to isomorphism. The presentation hS j Ri is finite if both sets S and R are finite. A group is finitely presented if it has at least one finite presentation. Presentations provide a universal method to describe groups. Example of finite presentations 1 G = hs1;:::; sn j [si ; sj ]; 81 ≤ i < j ≤ ni is the free abelian group of rank n.
    [Show full text]
  • Combinatorial Group Theory
    Combinatorial Group Theory Charles F. Miller III 7 March, 2004 Abstract An early version of these notes was prepared for use by the participants in the Workshop on Algebra, Geometry and Topology held at the Australian National University, 22 January to 9 February, 1996. They have subsequently been updated and expanded many times for use by students in the subject 620-421 Combinatorial Group Theory at the University of Melbourne. Copyright 1996-2004 by C. F. Miller III. Contents 1 Preliminaries 3 1.1 About groups . 3 1.2 About fundamental groups and covering spaces . 5 2 Free groups and presentations 11 2.1 Free groups . 12 2.2 Presentations by generators and relations . 16 2.3 Dehn’s fundamental problems . 19 2.4 Homomorphisms . 20 2.5 Presentations and fundamental groups . 22 2.6 Tietze transformations . 24 2.7 Extraction principles . 27 3 Construction of new groups 30 3.1 Direct products . 30 3.2 Free products . 32 3.3 Free products with amalgamation . 36 3.4 HNN extensions . 43 3.5 HNN related to amalgams . 48 3.6 Semi-direct products and wreath products . 50 4 Properties, embeddings and examples 53 4.1 Countable groups embed in 2-generator groups . 53 4.2 Non-finite presentability of subgroups . 56 4.3 Hopfian and residually finite groups . 58 4.4 Local and poly properties . 61 4.5 Finitely presented coherent by cyclic groups . 63 1 5 Subgroup Theory 68 5.1 Subgroups of Free Groups . 68 5.1.1 The general case . 68 5.1.2 Finitely generated subgroups of free groups .
    [Show full text]
  • Arxiv:1711.08410V2
    LATTICE ENVELOPES URI BADER, ALEX FURMAN, AND ROMAN SAUER Abstract. We introduce a class of countable groups by some abstract group- theoretic conditions. It includes linear groups with finite amenable radical and finitely generated residually finite groups with some non-vanishing ℓ2- Betti numbers that are not virtually a product of two infinite groups. Further, it includes acylindrically hyperbolic groups. For any group Γ in this class we determine the general structure of the possible lattice embeddings of Γ, i.e. of all compactly generated, locally compact groups that contain Γ as a lattice. This leads to a precise description of possible non-uniform lattice embeddings of groups in this class. Further applications include the determination of possi- ble lattice embeddings of fundamental groups of closed manifolds with pinched negative curvature. 1. Introduction 1.1. Motivation and background. Let G be a locally compact second countable group, hereafter denoted lcsc1. Such a group carries a left-invariant Radon measure unique up to scalar multiple, known as the Haar measure. A subgroup Γ < G is a lattice if it is discrete, and G/Γ carries a finite G-invariant measure; equivalently, if the Γ-action on G admits a Borel fundamental domain of finite Haar measure. If G/Γ is compact, one says that Γ is a uniform lattice, otherwise Γ is a non- uniform lattice. The inclusion Γ ֒→ G is called a lattice embedding. We shall also say that G is a lattice envelope of Γ. The classical examples of lattices come from geometry and arithmetic. Starting from a locally symmetric manifold M of finite volume, we obtain a lattice embedding π1(M) ֒→ Isom(M˜ ) of its fundamental group into the isometry group of its universal covering via the action by deck transformations.
    [Show full text]
  • Finitely Generated Subgroups of Lattices in Psl2 C
    FINITELY GENERATED SUBGROUPS OF LATTICES IN PSL2 C YAIR GLASNER, JUAN SOUTO, AND PETER STORM Abstract. Let Γ be a lattice in PSL2(C). The pro-normal topology on Γ is defined by taking all cosets of non-trivial normal subgroups as a basis. This topology is finer than the pro-finite topology, but it is not discrete. We prove that every finitely generated subgroup ∆ < Γ is closed in the pro-normal topology. As a corollary we deduce that if H is a maximal subgroup of a lattice in PSL2(C) then either H is finite index or H is not finitely generated. 1. Introduction In most infinite groups, it is virtually impossible to understand the lattice of all subgroups. Group theorists therefore focus their attention on special families of subgroups such as finite index subgroups, normal subgroups, or finitely generated subgroups. This paper will study the family of finitely generated subgroups of lattices in PSL2(C). Recall that lattices in PSL2(C) are the fundamental groups of finite volume hyperbolic 3-orbifolds. For these groups we establish a connection between finitely generated subgroups and normal subgroups. This connection is best expressed in the following topological terms. If a family of subgroups N is invariant under conjugation and satisfies the con- dition, (1) for all N1,N2 ∈ N ∃ N3 ∈ N such that N3 ≤ N1 ∩ N2 then one can define an invariant topology in which the given family of groups constitutes a basis of open neighborhoods for the identity element. The most famous example is the pro-finite topology, which is obtained by taking N to be the family of (normal) finite index subgroups.
    [Show full text]
  • Relative Subgroup Growth and Subgroup Distortion
    Relative Subgroup Growth and Subgroup Distortion Tara C. Davis, Alexander Yu. Olshanskii October 29, 2018 Abstract We study the relative growth of finitely generated subgroups in finitely generated groups, and the corresponding distortion function of the embed- dings. We explore which functions are equivalent to the relative growth functions and distortion functions of finitely generted subgroups. We also study the connections between these two asymptotic invariants of group embeddings. We give conditions under which a length function on a finitely generated group can be extended to a length function on a larger group. 1 Introduction and Background We will use the notation that for a group G with finite generating set X, and for an element g G, then g X represents the word length of ∈ | | the element g with respect to the generating set X. For this paper, N = 1, 2,..., . For r N we use the notation that BG(r) represents the ball { } ∈ in G centered at the identity with radius r. That is, BG(r) = g G : { ∈ g X r . | | ≤ } Definition 1.1. Let G be a group finitely generated by a set X with any subset H. The growth function of H in G is G g : N N : r # h H : h X r = #(BG(r) H). H → → { ∈ | | ≤ } ∩ arXiv:1212.5208v1 [math.GR] 20 Dec 2012 The growth functions of subsets and other functions of this type be- come asymptotic invariants independent of the choice of a finite set X if one takes them up to a certain equivalence (see the definitions of , , ∼ ∼ and in Section 3.) ≈ The growth of a subgroup H with respect to a finite set of generators of a bigger group G is called the relative growth of H in G and denoted by G G gH .
    [Show full text]
  • Myasnikov.1.Free.Groups.Pdf
    Free groups Contents 1 Free groups 1 1.1 De¯nitions and notations . 1 1.2 Construction of a free group with basis X ........... 2 1.3 The universal property of free groups. 6 1.4 Presentations of groups . 8 1.5 Rank of free groups . 9 1 Free groups 1.1 De¯nitions and notations Let G be a group. If H is a subgroup of G then we write H · G; if H is a normal subgroup of G we write H E G. For a subset A ⊆ G by hAi we denote the subgroup of G generated by A (the intersection of all subgroups of G containing A). It is easy to see that "1 "n hAi = fai1 ; : : : ; ain j aij 2 A; "j 2 f1; ¡1g; n 2 Ng "1 "n To treat expressions like ai1 ; : : : ; ain a little bit more formally we need some terminology. Let X be an arbitrary set. A word in X is a ¯nite sequence of elements (perhaps, empty) w which we write as w = y1 : : : yn ( yi 2 X). The number n is called the length of the word w, we denote it by jwj. We denote the empty word by ² and put j²j = 0. Consider X¡1 = fx¡1jx 2 Xg; where x¡1 is just a formal expression obtained from x and ¡1. If x 2 X then the symbols x and x¡1 are called literals in X. Denote by X§1 = X [ X¡1 1 2 the set of all literals in X. For a literal y 2 X§1 we de¯ne y¡1 as ½ x¡1; if y = x 2 X; y¡1 = x; if y = x¡1 2 X.
    [Show full text]
  • On Finitely Generated Profinite Groups, I 173
    Annals of Mathematics, 165 (2007), 171–238 On finitely generated profinite groups, I: strong completeness and uniform bounds By Nikolay Nikolov* and Dan Segal Abstract We prove that in every finitely generated profinite group, every subgroup of finite index is open; this implies that the topology on such groups is deter- mined by the algebraic structure. This is deduced from the main result about finite groups: let w be a ‘locally finite’ group word and d ∈ N. Then there exists f = f(w, d) such that in every d-generator finite group G, every element of the verbal subgroup w(G) is equal to a product of fw-values. An analogous theorem is proved for commutators; this implies that in every finitely generated profinite group, each term of the lower central series is closed. The proofs rely on some properties of the finite simple groups, to be established in Part II. Contents 1. Introduction 2. The Key Theorem 3. Variations on a theme 4. Proof of the Key Theorem 5. The first inequality: lifting generators 6. Exterior squares and quadratic maps 7. The second inequality, soluble case 8. Word combinatorics 9. Equations in semisimple groups, 1: the second inequality 10. Equations in semisimple groups, 2: powers 11. Equations in semisimple groups, 3: twisted commutators *Work done while the first author held a Golda-Meir Fellowship at the Hebrew University of Jerusalem. 172 NIKOLAY NIKOLOV AND DAN SEGAL 1. Introduction A profinite group G is the inverse limit of some inverse system of finite groups. Thus it is a compact, totally disconnected topological group; prop- erties of the original system of finite groups are reflected in properties of the topological group G.
    [Show full text]
  • 2.4 Direct Products and Finitely Generated Abelian Groups
    2.4. DIRECT PRODUCTS AND FINITELY GENERATED ABELIAN GROUPS31 2.4 Direct Products and Finitely Generated Abelian Groups 2.4.1 Direct Products 4 Def 2.43. The Cartesian product of sets S1,S2, ··· ,Sn is the set of all ordered n-tuples (a1, ··· , an) where ai ∈ Si for i = 1, 2 ··· , n. n Y Si = S1 × S2 × · · · × Sn i=1 := {(a1, a2, ··· , an) | ai ∈ Si for i = 1, 2, ··· , n}. Caution: The notation (a1, a2, ··· , an) here has different meaning from the cycle notation in permutation groups. Thm 2.44. If G1,G2, ··· ,Gn are groups with the corresponding multiplica- Qn tions, then i=1 Gi is a group under the following multiplication: (a1, a2, ··· , an)(b1, b2, ··· , bn) := (a1b1, a2b2, ··· , anbn) Qn for (a1, a2, ··· , an) and (b1, b2, ··· , bn) in i=1 Gi. Proof. Qn 1. Closed: ai ∈ Gi, bi ∈ Gi ⇒ aibi ∈ Gi ⇒ (a1b1, ··· , anbn) ∈ i=1 Gi. 2. Associativity: By the associativity in each Gi. 3. Identity: Let ei be the identity in Gi. Then e := (e1, e2, ··· , en) is the identity of G. −1 −1 4. Inverse: The inverse of (a1, ··· , an) is (a1 , ··· , an ). Ex 2.45. R2, R3 are abelian groups. (Every finite dimensional real vector space is an abelian group that is iso- morphic to certain Rn.) Ex 2.46. Z2 = {(a, b) | a, b ∈ Z} is a subgroup of R2. Qn Qn (If Hi ≤ Gi for i = 1, ··· , n, then i=1 Hi ≤ i=1 Gi). 41st HW: 1, 2, 14, 34, 46 32 CHAPTER 2. PERMUTATIONS, COSETS, DIRECT PRODUCTS Ex 2.47.
    [Show full text]