DOCSLIB.ORG

Polycyclic Groups

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Polycyclic Groups Polycyclic groups Eamonn O'Brien September 2010 Introduction Let F be free group on a non-empty set X. A group presentation is a set consisting of X and a set, R, of words in X. If R is the normal closure of R in F , the group G defined by the presen- tation is F=R and is written h X : R i. Example 1. G = ha; bja4; b2; ab = a−1i H = ha; bja4; b2 = a2; ab = a−1i What can we discover about the structure of G or H? One area of substantial progress at algorithmic and computational level is in the study of particular quotients of G. Examples include abelian, p-quotient, soluble quotients. May discover that G infinite, by examining the invariants of its largest abelian quotient. Can compute \useful" presentations for other quotients of the group: those which have prime-power order, are nilpotent, or are soluble. Central feature of these presentations is that they provide a solution to the word problem: Decide if two words in the generators X represent the same element of G. Outline of lecture series Abelian quotients. Polycyclic generating sequences: basic properties. 1 Polycyclic presentations: consistency and collection. Constructing polycyclic presentations. Generating descriptions of p-groups. An application: SmallGroups. Constructing automorphism groups of p-groups. Deciding isomorphism of p-groups. 2 Abelian quotients Lemma 2. G=N abelian if and only if N ≥ G0. Largest abelian quotient of G is G=G0. Structure of this abelian group can be determined fairly readily. Definition 3. A is in Smith Normal Form if for some k ≥ 0 the entries di = Ai;i for 1 ≤ i ≤ k are positive, A has no other non-zero entries, and dijdi+1 for 1 ≤ i ≤ k. Determine the structure of G=G0 1. Abelianise the presentation of G by adding relations to make G abelian. 0 ∼ n n 2. G=G = Z =B where B is a subgroup of Z . 3. Describe B by a matrix S(B). n 4. To obtain the structure of Z =B, we apply row-and-column operations to S(B) to convert it to Smith normal form S. n 5. We read off abelian invariants of Z =B from S. Example 4. G = hx; y; zj(xyz−1)2; (x−1y2z)2; (xy−2z−1)2i Abelianise to obtain 2 G=G0 = hx; y; zj(xyz−1)2; (x−1y2z)2; (xy−2z−1) ; xy = yx; xz = zx; yz = zyi 0 2 2 −21 Describe B by S(B) = @−2 4 2 A 2 −4 −2 02 0 01 Smith Normal form of S(B) is @0 6 0A 0 0 0 0 ∼ Hence G=G = Z2 × Z6 × Z and so it is infinite. 3 Polycyclic Groups Definition 5. G is polycyclic if it has a descending chain of subgroups G = G1 ≥ G2 ≥ · · · ≥ Gn+1 = 1 in which Gi+1 C Gi, and Gi=Gi+1 is cyclic. Such a chain of subgroups is called a polycyclic series. Polycyclic groups: solvable groups in which every subgroup is finitely generated. Example 6. G = Alt(4) = h(1; 3)(2; 4); (1; 2)(3; 4); (1; 2; 3)i where V = h(1; 3)(2; 4); (1; 2)(3; 4)iC G and Z2 = h(1; 3)(2; 4)i C V . So Alt(4) B V B Z2. Polycyclic sequences Let G be polycyclic with polycyclic series G = G1 ≥ G2 ≥ · · · ≥ Gn+1 = 1. Since Gi=Gi+1 is cyclic, there exist xi 2 G with hxiGi+1i = Gi=Gi+1 for every i 2 f1; : : : ; ng. Definition 7. A sequence of elements X = [x1; : : : ; xn] such that hxiGi+1i = Gi=Gi+1 for 1 ≤ i ≤ n is a polycyclic sequence (PCGS) for G. Definition 8. Let X be a PCGS sequence for G. The sequence R(X) := (r1; : : : ; rn) defined by ri := jGi :Gi+1j 2 N [ f1g is the sequence of relative orders for X. Let I(X) := fi 2 f1 : : : ng j ri finiteg. Example 9. X := [(1; 2; 3); (1; 2)(3; 4); (1; 3)(2; 4)] is PCGS for Alt(4) where R(X) = [3; 2; 2] and I(X) = [1; 2; 3]. Relative orders exhibit information about G. G is finite iff every entry in R(X) is finite or, equivalently iff I(X) = f1 : : : ng. If G is finite, then jGj = r1 ··· rn, the product of the entries in R(X). ∼ Example 10. Let G := h(1; 2; 3; 4); (1; 3)i = D8. ∼ a) Let G2 := h(1; 2; 3; 4)i = C4. Then G = G1 ≥ G2 ≥ G3 = 1 is polycyclic series for G. X := [(1; 3); (1; 2; 3; 4)] and Y := [(2; 4); (1; 4; 3; 2)] are PCGS defining this series. R(X) = R(Y ) = (2; 4) and I(X) = I(Y ) = f1; 2g. 4 ∼ ∼ b) Let G2 := h(1; 2)(3; 4); (1; 3)(2; 4)i = V and G3 := h(1; 3)(2; 4)i = C2. So G = G1 ≥ G2 ≥ G3 ≥ G4 = 1. X := [(2; 4); (1; 2)(3; 4); (1; 3)(2; 4)] and Y := [(1; 2; 3; 4); (1; 2)(3; 4); (1; 3)(2; 4)] are polycyclic sequences defining this series. R(X) = R(Y ) = (2; 2; 2) and I(X) = I(Y ) = f1; 2; 3g. Example 11. Let G := ha; bi with −1 0 −1 −1 a := and b := : 0 1 0 1 ∼ G = D1, the infinite dihedral group. A polycyclic sequence for G is X := [a; ab] with relative orders R(X) = (2; 1) and finite orders I(X) = f1g. Lemma 12. Let X = [x1; : : : ; xn] be a polycyclic sequence for G with the relative orders R(X) = (r1; : : : ; rn). For every g 2 G there exists a sequence (e1; : : : ; en), with ei 2 Z for 1 ≤ i ≤ n and 0 ≤ ei < ri if i 2 I(X), such that e1 en g = x1 ··· xn . e1 Proof. Since G1=G2 = hx1G2i, we find that gG2 = x1 G2 for some e1 2 Z. If 1 2 I(X), then r1 < 1 and we can choose ei 2 f0 : : : r1 −1g. −e1 Let h = x1 g 2 G2. By induction on the length of a polycyclic sequence, we can assume that e2 en we know expression of the desired form for h; that is, h = x2 ··· xn . e1 e2 en Hence g = x1 x2 ··· xn as desired. Example 13. G = Alt(4) X := [x1 = (1; 2; 3); x2 = (1; 2)(3; 4); x3 = (1; 3)(2; 4)] is PCGS for G where R(X) = [3; 2; 2] and I(X) = [1; 2; 3]. V = hx2; x3i and H = hx3i. g = (1; 2; 4). 2 −2 gV = x1V so x1 g = (1; 4)(2; 3) 2 V . −1 Now v := (1; 4)(2; 3) satisfies vH = x2H, so x2 v = (1; 3)(2; 4) 2 H. −1 Hence x2 v = x3 so v = x2x3. 2 Hence g = x1x2x3. 5 Normal form e1 en Definition 14. The expression g = x1 ··· xn is the normal form of g 2 G with respect to X. The sequence expX (g) := (e1; : : : ; en) is the exponent vector of g with respect to X. n Can define an injective map G ! Z : g 7! expX (g) from G into the n additive group of Z . This is not a group homomorphism! Polycyclic group to presentation? Exponent vectors of elements of G can be used to describe relations for G in terms of X. Lemma 15. Let X = [x1; : : : ; xn] be a polycyclic sequence for G with relative orders R(X) = (r1; : : : ; rn). ri ri ai;i+1 ai;n a) Let i 2 I(X). The normal form of a power xi is xi = xi+1 ··· xn . −1 b) Let 1 ≤ j < i ≤ n. The normal form of a conjugate xj xixj is −1 bi;j;j+1 bi;j;n xj xixj = xj+1 ··· xn . −1 c) Let 1 ≤ j < i ≤ n. The normal form of a conjugate xjxixj is −1 ci;j;j+1 ci;j;n xjxixj = xj+1 ··· xn . Polycyclic presentation Definition 16. A presentation f x1; : : : ; xn j R g is a polycyclic presentation if there is a sequence S = (s1; : : : ; sn) with si 2 N [ f1g and integers ai;k; bi;j;k; ci;j;k such that R consists of the following relations: si ai;i+1 ai;n xi = xi+1 ··· xn for 1 ≤ i ≤ n with si < 1; −1 bi;j;j+1 bi;j;n xj xixj = xj+1 ··· xn for 1 ≤ j < i ≤ n; −1 ci;j;j+1 ci;j;n xjxixj = xj+1 ··· xn for 1 ≤ j < i ≤ n: We describe the presentation by Pch x1; : : : ; xn j R i. If G is defined by such a polycyclic presentation then G is a PC-group. Theorem 17. Every polycyclic sequence determines a (unique) polycyclic presentation. Thus every polycyclic group can be defined by a polycyclic presentation. 6 Example 18. Let D8 := h(1; 3); (1; 2; 3; 4)i with polycyclic sequence X := [(1; 3); (1; 2; 3; 4)] and relative orders R(X) = (2; 4). Polycyclic presentation defined by X has generators x1; x2, power expo- 2 4 −1 3 nents s1 = 2 and s2 = 4. Relations are x1 = 1, x2 = 1, x1x2x1 = x2 and −1 3 x1 x2x1 = x2. Example 19. S4 has PCGS X = [(3; 4); (2; 4; 3); (1; 3)(2; 4); (1; 2)(3; 4)] where R(X) = [2; 3; 2; 2]. 2 3 2 2 x1 2 Pchx1; x2; x3; x4 j x1 = x2 = x3 = x4 = 1; x2 = x2; x1 x2 x2 x3 = x3x4; x3 = x4; x4 = x3x4i Finite p-groups Usually write power-commutator presentation.
Load more

Recommended publications
  • Polynomial Structures on Polycyclic Groups
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 349, Number 9, September 1997, Pages 3597{3610 S 0002-9947(97)01924-7 POLYNOMIAL STRUCTURES ON POLYCYCLIC GROUPS KAREL DEKIMPE AND PAUL IGODT Abstract. We know, by recent work of Benoist and of Burde & Grunewald, that there exist polycyclic–by–finite groups G,ofrankh(the examples given were in fact nilpotent), admitting no properly discontinuous affine action on Rh. On the other hand, for such G, it is always possible to construct a prop- erly discontinuous smooth action of G on Rh. Our main result is that any polycyclic–by–finite group G of rank h contains a subgroup of finite index act- ing properly discontinuously and by polynomial diffeomorphisms of bounded degree on Rh. Moreover, these polynomial representations always appear to contain pure translations and are extendable to a smooth action of the whole group G. 1. Introduction. In 1977 ([18]), John Milnor asked if it was true that any torsion-free polycyclic– by–finite group G occurs as the fundamental group of a compact, complete, affinely flat manifold. This is equivalent to saying that G acts properly discontinuously and h G via affine motions on R ( ). (Throughout this paper, we will use h(G)todenotethe Hirsch length of a polycyclic–by–finite group G.) As there are several definitions of the notion “properly discontinuous action” we write down the definition we use: Definition 1.1. Let Γ be a group acting (on the left) on a Hausdorff space X.We say that the action of Γ on X is properly discontinuous if and only if γ 1.
    [Show full text]
  • Group and Ring Theoretic Properties of Polycyclic Groups Algebra and Applications
    Group and Ring Theoretic Properties of Polycyclic Groups Algebra and Applications Volume 10 Managing Editor: Alain Verschoren University of Antwerp, Belgium Series Editors: Alice Fialowski Eötvös Loránd University, Hungary Eric Friedlander Northwestern University, USA John Greenlees Sheffield University, UK Gerhard Hiss Aachen University, Germany Ieke Moerdijk Utrecht University, The Netherlands Idun Reiten Norwegian University of Science and Technology, Norway Christoph Schweigert Hamburg University, Germany Mina Teicher Bar-llan University, Israel Algebra and Applications aims to publish well written and carefully refereed mono- graphs with up-to-date information about progress in all fields of algebra, its clas- sical impact on commutative and noncommutative algebraic and differential geom- etry, K-theory and algebraic topology, as well as applications in related domains, such as number theory, homotopy and (co)homology theory, physics and discrete mathematics. Particular emphasis will be put on state-of-the-art topics such as rings of differential operators, Lie algebras and super-algebras, group rings and algebras, C∗-algebras, Kac-Moody theory, arithmetic algebraic geometry, Hopf algebras and quantum groups, as well as their applications. In addition, Algebra and Applications will also publish monographs dedicated to computational aspects of these topics as well as algebraic and geometric methods in computer science. B.A.F. Wehrfritz Group and Ring Theoretic Properties of Polycyclic Groups B.A.F. Wehrfritz School of Mathematical Sciences
    [Show full text]
  • Abelian-By-Polycyclic Groups
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 41, Number 2, December 1973 FINITE FRATTINI FACTORS IN FINITELY GENERATED ABELIAN-BY-POLYCYCLIC GROUPS JOHN C. LENNOX Abstract. A criterion for a group to be finite is used to prove that subgroups of finitely generated Abelian-by-polycyclic groups are finite if their Frattini factor groups are finite. In a previous paper [2] we considered finitely generated soluble groups and their subgroups in relation to the question whether they were finite if their Frattini factor groups were finite. Among other results we answered this question in the affirmative for subgroups of finitely generated Abelian- by-nilpotent groups [2, Theorem B]. The class of all finitely generated Abelian-by-nilpotent groups is of course a proper subclass of the class of all finitely generated Abelian-by-polycyclic groups, since finitely generated nilpotent groups are polycyclic, but the converse does not hold. Now, although our methods in [2] depended on properties of finitely generated Abelian-by-polycyclic groups (more accurately, they depended on prop- erties of Hall's class 23 of Abelian groups where B is in 23 if and only if B is an Abelian group which can be extended to a finitely generated group G such that G/B is polycyclic), they were not good enough to settle the question for the wider class of groups. Our object here is to close this gap by proving the Theorem. Subgroups of finitely generated Abelian-by-polycyclic groups are finite if their Frattini factor groups are finite. The information we need here, supplementary to that contained in [2], is given in the following result which generalises Lemma 3 of [2] in a way which may be of interest in itself.
    [Show full text]
  • Polycyclic, Metabelian Or Soluble of Type (FP) $ {\Infty} $ Groups With
    POLYCYCLIC, METABELIAN OR SOLUBLE OF TYPE (FP)∞ GROUPS WITH BOOLEAN ALGEBRA OF RATIONAL SETS AND BIAUTOMATIC SOLUBLE GROUPS ARE VIRTUALLY ABELIAN VITALY ROMAN’KOV Omsk State Dostoevsky University and Omsk State Technical University, 644077, Omsk, Russia, e-mail:[email protected] Abstract Let G be a polycyclic, metabelian or soluble of type (FP)∞ group such that the class Rat(G) of all rational subsets of G is a boolean algebra. Then G is virtually abelian. Every soluble biautomatic group is virtually abelian. 1 Introduction The topic of this paper is two important concepts: rational sets and biauto- matic structure. We study finitely generated soluble groups G such that the class Rat(G) of all rational subsets of G is a boolean algebra. We conjecture that every such group is virtually abelian. Note that every finitely generated virtually abelian group satisfies to this property. We confirm this conjecture in the case where G is a polycyclic, metabelian or soluble group of type FP∞. This conjecture remains open in general case. It appeared that the notion FP∞ helps to prove by the way that every soluble biautomatic group is virtually abelian. Thus, we give answer to known question posed in [1]. We provide full proofs of four theorems attributed to Bazhenova, a former student of the author, stating that some natural assumptions imply that a sol- uble group is virtually abelian. The original proofs was given by her in collab- arXiv:1511.00223v1 [math.GR] 1 Nov 2015 oration with the author more than 14 years ago and had never been published.
    [Show full text]
  • Solvable Groups Acting on the Line 403
    TRANSACTIONS of the AMERICAN MATHEMATICAL SOCIETY Volume 278, Number I. July 1983 SOLVABLEGROUPS ACTING ON THE LINE BY J. F. PLANTE1 Abstract. Actions of discrete groups on the real line are considered. When the group of homeomorphisms is solvable several sufficient conditions are given which guarantee that the group is semiconjugate to a subgroup of the affine group of the line. In the process of obtaining these results sufficient conditions are also de- termined for the existence of invariant (quasi-invariant) measures for abelian (solva- ble) groups acting on the line. It turns out, for example, that any solvable group of real analytic diffeomorphisms or a polycyclic group of homeomorphisms has a quasi-invariant measure, and that any abelian group of C diffeomorphisms has an invariant measure. An example is given to show that some restrictions are necessary in order to obtain such conclusions. Some applications to the study of codimension one foliations are indicated. Introduction. If G is a group of homeomorphisms of the line it is reasonable to ask for a dynamic description of how G acts when reasonable restrictions are placed on G. For example, in [9] it is shown that if G is finitely generated and nilpotent then there is a G-invariant Borel measure on R which is finite on compact sets. This already says much about G. The proof is quite different from classical arguments which guarantee the existence of a finite invariant measure when an amenable group acts on a compact Hausdorff space [6] in that one uses pseudogroup properties of G rather than group properties (specifically, nonexponential growth of orbits rather than nilpotence of G).
    [Show full text]
  • New York Journal of Mathematics Dynamics of Flat Actions on Totally Disconnected, Locally Compact Groups
    New York Journal of Mathematics New York J. Math. 22 (2016) 115{190. Dynamics of flat actions on totally disconnected, locally compact groups Colin D. Reid Abstract. Let G be a totally disconnected, locally compact group and let H be a virtually flat (for example, polycyclic) group of automor- phisms of G. We study the structure of, and relationships between, various subgroups of G defined by the dynamics of H. In particular, we consider the following four subgroups: the intersection of all tidy subgroups for H on G (in the case that H is flat); the intersection of all H-invariant open subgroups of G; the smallest closed H-invariant subgroup D such that no H-orbit on G=D accumulates at the trivial coset; and the group generated by the closures of contraction groups of elements of H on G. Contents 1. Introduction 116 2. Preliminaries 127 2.1. Tidy theory for cyclic actions 128 2.2. Flat groups 132 2.3. Metrizability 135 3. The relative Tits core 136 3.1. Prior results on stability of the contraction group 136 3.2. Invariance of contraction groups and the relative Tits core137 3.3. Relative Tits cores and elementary groups 140 3.4. Subgroups containing relative Tits cores 144 3.5. Examples 145 4. The nub of a flat group 146 4.1. Introduction 146 4.2. Invariant uniscalar subgroups 148 4.3. Tidy subgroups in quotients 151 4.4. Flatness below 154 Received April 4, 2015. 2010 Mathematics Subject Classification. 22D05. Key words and phrases. T.d.l.c.
    [Show full text]
  • On the Fitting Subgroup of a Polycylic-By-Finite Group and Its Applications
    Journal of Algebra 242, 176᎐187Ž. 2001 doi:10.1006rjabr.2001.8803, available online at http:rrwww.idealibrary.com on On the Fitting Subgroup of a Polycyclic-by-Finite Group and Its Applications Bettina Eick View metadata, citation and similar papers at core.ac.uk brought to you by CORE ¨ Fachbereich fur¨¨ Mathematik und Informatik, Uni ersitat Kassel, provided by Elsevier - Publisher Connector 34109 Kassel, Germany E-mail: [email protected] Communicated by Walter Feit Received May 12, 2000 We present an algorithm for determining the Fitting subgroup of a polycyclic- by-finite group. As applications we describe methods for calculating the centre and the FC-centre and for exhibiting the nilpotent-by-abelian-by-finite structure of a polycyclic-by-finite group. ᮊ 2001 Academic Press 1. INTRODUCTION The Fitting subgroup of a polycyclic-by-finite group can be defined as its maximal nilpotent normal subgroup. For Fitting subgroups of finite groups various other characterizations are known: For example, they can be described as the centralizer of a chief series. We observe that the Fitting subgroup of a polycyclic-by-finite group can also be characterized as the centralizer of a certain type of series, and this series can be considered as a ‘‘generalized chief series.’’ We describe a practical algorithm for computing a generalized chief series of a polycyclic-by-finite group G. Our method is based on the determination of radicals of finite-dimensional KG-modules, where K is either a finite field or the rational numbers. Once a generalized chief series of G is given, we can construct its centralizer and thus we obtain an algorithm to compute the Fitting subgroup of G.
    [Show full text]
  • Arxiv:1302.5915V4 [Math.GR] 30 Apr 2015 BTATCMESRTR FLTIE NLIE in LATTICES of COMMENSURATORS ABSTRACT .Introduction 1
    ABSTRACT COMMENSURATORS OF LATTICES IN LIE GROUPS DANIEL STUDENMUND Abstract. Let Γ be a lattice in a simply-connected solvable Lie group. We construct a Q-defined algebraic group A such that the abstract commensurator of Γ is isomorphic to A(Q) and Aut(Γ) is commensu- rable with A(Z). Our proof uses the algebraic hull construction, due to Mostow, to define an algebraic group H so that commensurations of Γ extend to Q-defined automorphisms of H. We prove an analogous re- sult for lattices in connected linear Lie groups whose semisimple quotient satisfies superrigidity. Contents 1. Introduction 2 2. Notation and preliminaries 6 2.1. Algebraic groups 6 2.2. Semisimple Lie and algebraic groups 8 3. The abstract commensurator 9 4. Commensurations of lattices in nilpotent groups 11 4.1. Example: the Heisenberg group 11 4.2. Commensuristic subgroups 12 4.3. Commensurations are rational 13 5. The algebraic hull of a polycyclic group 13 5.1. Polycyclic groups 13 5.2. Algebraic hulls 14 5.3. Unipotent shadow 16 arXiv:1302.5915v4 [math.GR] 30 Apr 2015 5.4. Algebraic structure of Aut(H) 17 5.5. A finite index subgroup of Comm(Γ) 17 6. Commensurations of lattices in solvable groups 18 6.1. Example: Sol lattice 18 6.2. Commensurations of solvable lattices are rational 19 7. Commensurations of lattices in semisimple groups 22 7.1. Arithmetic lattices in semisimple groups 22 7.2. More general lattices in semisimple groups 24 7.3. Example: PGLn(Z) 25 8. Commensurations of general lattices 26 References 34 Date: June 24, 2018.
    [Show full text]
  • Matrix Groups
    Matrix groups Peter J. Cameron 1 Matrix groups and group representations These two topics are closely related. Here we consider some particular groups which arise most naturally as matrix groups or quotients of them, and special properties of matrix groups which are not shared by arbitrary groups. In repre- sentation theory, we consider what we learn about a group by considering all its homomorphisms to matrix groups. This article falls roughly into two parts. In the first part we discuss proper- ties of specific matrix groups, especially the general linear group (consisting of all invertible matrices of given size over a given field) and the related “classical groups”. In the second part, we consider what we learn about a group if we know that it is a linear group. Most group theoretic terminology is standard and can be found in any text- book. A few terms we need are summarised in the next definition. Let X and Y be group-theoretic properties. We say that a group G is locally X if every finite subset of G is contained in a subgroup with property X; G is X- by-Y if G has a normal subgroup N such that N has X and G/N has Y; and G is poly-X if G has subgroups N0 = 1,N1,...,Nr = G such that, for i = 0,...,r − 1, Ni is normal in Ni+1 and the quotient has X. Thus a group is locally finite if and only if every finitely generated subgroup is finite; and a group is solvable if and only if it is poly-abelian.
    [Show full text]
  • Infinite Approximate Subgroups of Soluble Lie Groups
    Mathematische Annalen https://doi.org/10.1007/s00208-021-02258-8 Mathematische Annalen Infinite approximate subgroups of soluble Lie groups Simon Machado1 Received: 12 November 2019 / Revised: 9 June 2021 / Accepted: 13 August 2021 © The Author(s) 2021 Abstract We study infinite approximate subgroups of soluble Lie groups. We show that approx- imate subgroups are close, in a sense to be defined, to genuine connected subgroups. Building upon this result we prove a structure theorem for approximate lattices in soluble Lie groups. This extends to soluble Lie groups a theorem about quasi-crystals due to Yves Meyer. Keywords Approximate lattices · Approximate subgroups · Lie groups · Linear algebraic groups Mathematics Subject Classification 05E15 · 22E99 · 22E40 1 Introduction Approximate subgroups were defined by Terence Tao [21] in order to give a non- commutative generalisation of results from additive combinatorics. On the one hand, finite approximate subgroups have been extensively studied. We can mention, among others, Gregory Freiman [7], Harald Helfgott [9], Ehud Hrushovski [10], László Pyber and Endre Szabó [16], Imre Ruzsa [18] and Emmanuel Breuillard, Ben Green and Terence Tao [4]. This lead to the structure theorem [4] that asserts that finite approx- imate subgroups are commensurable to coset nilprogressions, which are a certain non-commutative generalisation of arithmetic progressions. On the other hand, it Communicated by Andreas Thom. This work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L016516/1 for the University of Cambridge Centre for Doctoral Training, the Cambridge Centre for Analysis. B Simon Machado [email protected] 1 Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK 123 S.
    [Show full text]
  • Finite Groups*
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 142, 118-149 (1991) The Algorithmic Theory of Polycyclic-by- Finite Groups* GILBERT BAUMSLAG Department of Mathematics, City College, The City University of New York, New York, New York 10021 FRANK B. CANNONITO Department of Mathematics, University of California, Irvine, California 92664 DEREK J. S. ROBINSON Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801 AND DAN SEGAL AN Souls College, Oxford, England Communicated by Marshall Hall, Jr. Received October 1989 1. INTRODUCTION A group is said to be polycyclic-by-finite, or a PF-group for short, if it has a polycyclic normal subgroup of finite index. Equivalently, PF-groups are exactly the groups which have a series of finite length whose infinite factors are cyclic. By a well-known theorem of P. Hall every PF-group is finitely presented-and in fact PF-groups form the largest known section- closed class of finitely presented groups. It is this fact that makes PF-groups natural objects of study from the algorithmic standpoint. The general aim of the algorithmic theory of PF-groups can be described as the collection of information about a PF-group which can, in principle * During the final stage of work on this article the authors were visitors at the Mathemati- cal Sciences Research Institute in Berkeley. 118 0021~8693/91$3.00 Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved. POLYCYCLIC-BY-FINITE GROUPS 119 at least, be obtained by machine computation.
    [Show full text]
  • On the Nilpotent Length of Polycyclic Groups
    JOURNAL OF ALGEBRA 203, 125]133Ž. 1998 ARTICLE NO. JA977321 On the Nilpotent Length of Polycyclic Groups Gerard Endimioni* C.M.I., Uni¨ersite de Pro¨ence, UMR-CNRS 6632, 39, rue F. Joliot-Curie, 13453 Marseille Cedex 13, France Communicated by J. Tits Received October 30, 1996 Let G be a polycyclic group. We prove that if the nilpotent length of each finite quotient of G is bounded by a fixed integer n, then the nilpotent length of G is at most n. The case n s 1 is a well-known result of Hirsch. As a consequence, we obtain that if the nilpotent length of each 2-generator subgroup is at most n, then the nilpotent length of G is at most n. A more precise result in the case n s 2 permits us to prove that if each 3-generator subgroup is abelian-by-nilpotent, then G is abelian-by-nilpotent. Furthermore, we show that the nilpotent length of G equals the nilpotent length of the quotient of G by its Frattini subgroup. Q 1998 Academic Press 1. INTRODUCTION AND RESULTS We say that a group G is n-step nilpotent if it has a subnormal series Ä41sG01eGe??? eGns G of length n with each quotient nilpotent. In fact, substituting the core of Ž. Giiin G for G , we can assume that Giis0,1,...,n is a normal series. The least integer n such that G is n-step nilpotent is the nilpotent length of G. First, we shall prove in this paper the following THEOREM 1.
    [Show full text]