Locally Finite and Solvable Subgroups of Sfields
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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/. -
Base Size of Finite Primitive Solvable Permutation Groups
Base size of finite primitive solvable permutation groups Ayan Maiti Under the supervision of Prof. Pal Hegedus A thesis presented for the partial fulfilment towards the degree of Masters of Science in Mathematics Mathematics and its Application Central European University Hungary Declaration of Authorship I, Ayan Maiti, declare that this thesis titled, "Base size of finite primitive solvable per- mutation groups" and the work presented in it are my own. I confirm that: This work was done wholly or mainly while in candidature for a masters degree at this University. Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated. Where I have consulted the published work of others, this is always clearly attributed. Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. I have acknowledged all main sources of help. Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself. Signed: Ayan Maiti Date: 20th May, 2016 1 Abstract The content of this thesis report is based on the bounds of the base size of affine type primitive permutation groups, the bound was conjectured by Pyber and later was proved by Akos Seress. The the primary focus of this thesis is to understand the basic idea and the proof given by Akos Seress. -
On the Centralizer of an Element of Order Four in a Locally Finite Group
Glasgow Math. J. 49 (2007) 411–415. C 2007 Glasgow Mathematical Journal Trust. doi:10.1017/S0017089507003746. Printed in the United Kingdom ON THE CENTRALIZER OF AN ELEMENT OF ORDER FOUR IN A LOCALLY FINITE GROUP PAVEL SHUMYATSKY Department of Mathematics, University of Brasilia, Brasilia-DF, 70910-900 Brazil e-mail: [email protected] (Received 12 December, 2006; revised 6 March, 2007; accepted 10 March, 2007) Abstract. We prove that if G is a locally finite group admitting an automorphism φ of order four such that CG(φ) is Chernikov, then G has a soluble subgroup of finite index. 1. Introduction. The study of centralizers in locally finite groups is an interesting direction of group theory. In particular, the situation that a locally finite group G admits an automorphism φ of finite order (or, equivalenly, contains an element φ) such that CG(φ) is Chernikov has received some attention in the past. Recall that a group C is Chernikov if it has a subgroup of finite index that is a direct product of finitely many groups of type Cp∞ for various primes p (quasicyclic p-groups). By a deep result obtained independently by Shunkov [22] and also by Kegel and Wehrfritz [7] Chernikov groups are precisely the locally finite groups satisfying the minimal condition on subgroups; that is, any non-empty set of subgroups possesses a minimal subgroup. Hartley proved in 1988 that if φ is of prime-power order, then G has a locally soluble subgroup of finite index (see [4, Theorem 1]). The case of the theorem where φ has order two was handled by Asar in [1] and for automorphisms of arbitrary prime order the theorem was proved by Turau in [25]. -
Groups with Bounded Centralizer Chains and The~ Borovik--Khukhro
Groups with bounded centralizer chains and the Borovik–Khukhro conjecture A. A. Buturlakin1,2,3, D. O. Revin1,2, A. V. Vasil′ev1,2 Abstract Let G be a locally finite group and F (G) the Hirsch–Plotkin radical of G. Denote by S the full inverse image of the generalized Fitting subgroup of G/F (G) in G. Assume that there is a number k such that the length of every chain of nested centralizers in G does not exceed k. The Borovik–Khukhro conjecture states, in particular, that under this assumption the quotient G/S contains an abelian subgroup of index bounded in terms of k. We disprove this statement and prove some its weaker analog. Keywords: locally finite groups, centralizer lattice, c-dimension, k-tuple Dick- son’s conjecture Introduction Following [24], the maximal length of chains of nested centralizers of a group G is called the c-dimension of G (the same number is also known as the length of the centralizer lattice of G, see, e.g., [25]). Here we consider groups of finite c-dimension. The class of such groups is quite wide: it includes, for example, abelian groups, linear groups, torsion-free hyperbolic groups, and hence free groups. It is closed under taking subgroups, finite direct products, and extensions by finite groups; however, the c-dimension of a homomorphic image of a group from this class can be infinite. Obviously, each group of finite c-dimension satisfies the minimal condition on centralizers, therefore, it is a so-called MC -group. The minimal condition on centralizers is a useful notion in the class of arXiv:1708.03057v1 [math.GR] 10 Aug 2017 locally finite groups. -
Global Rigidity of Solvable Group Actions on S1 Lizzie Burslem Amie Wilkinson
ISSN 1364-0380 (on line) 1465-3060 (printed) 877 eometry & opology G T Volume 8 (2004) 877–924 Published: 5 June 2004 Global rigidity of solvable group actions on S1 Lizzie Burslem Amie Wilkinson Department of Mathematics, University of Michigan 2074 East Hall, Ann Arbor, MI 48109-1109 USA and Department of Mathematics, Northwestern University 2033 Sheridan Road, Evanston, IL 60208-2730 USA Email: [email protected], [email protected] Abstract In this paper we find all solvable subgroups of Diffω(S1) and classify their actions. We also investigate the Cr local rigidity of actions of the solvable Baumslag–Solitar groups on the circle. The investigation leads to two novel phenomena in the study of infinite group actions on compact manifolds. We exhibit a finitely generated group Γ and a manifold M such that: Γ has exactly countably infinitely many effective real-analytic actions on • M , up to conjugacy in Diffω(M); every effective, real analytic action of Γ on M is Cr locally rigid, for • some r 3, and for every such r, there are infinitely many nonconjugate, ≥ effective real-analytic actions of Γ on M that are Cr locally rigid, but not Cr−1 locally rigid. AMS Classification numbers Primary: 58E40, 22F05 Secondary: 20F16, 57M60 Keywords: Group action, solvable group, rigidity, Diffω(S1) Proposed:RobionKirby Received:26January2004 Seconded: Martin Bridson, Steven Ferry Accepted: 28 May 2004 c eometry & opology ublications G T P 878 Lizzie Burslem and Amie Wilkinson Introduction This paper describes two novel phenomena in the study of infinite group actions on compact manifolds. We exhibit a finitely generated group Γ and a manifold M such that: Γ has exactly countably infinitely many effective real-analytic actions on • M , up to conjugacy in Diffω(M); every effective, real analytic action of Γ on M is Cr locally rigid, for • some r 3, and for every such r, there are infinitely many nonconjugate, ≥ effective real-analytic actions of Γ on M that are Cr locally rigid, but not Cr−1 locally rigid. -
Uniformly Recurrent Subgroups of Simple Locally Finite Groups
UNIFORMLY RECURRENT SUBGROUPS OF SIMPLE LOCALLY FINITE GROUPS SIMON THOMAS Abstract. We study the uniformly recurrent subgroups of simple locally finite groups. 1. Introduction Let G be a countably infinite group and let SubG be the compact space of subgroups H 6 G. Then G acts as a group of homeomorphisms of SubG via the g −1 conjugation action, H 7! gHg . Following Glasner-Weiss [5], a subset X ⊆ SubG is said to be a uniformly recurrent subgroup or URS if X is a minimal G-invariant closed subset of SubG. For example, if N E G is a normal subgroup, then the singleton set f N g is a URS of G. Throughout this paper, these singleton URSs will be regarded as trivial URSs. More interesting examples of URSs arise as the stabilizer URSs of minimal actions. For example, suppose that ∆ is a compact space and that G y ∆ is a minimal G-action. Let f : ∆ ! SubG be the G-equivariant stabilizer map defined by x 7! Gx = f g 2 G j g · x = x g: and let X∆ = f(∆). If f is continuous, then it follows easily that X∆ is a URS of G; and, as expected, X∆ is called the stabilizer URS of the minimal action G y ∆. (It is well-known and easily checked that the map f : ∆ ! SubG is continuous if and only Fix∆(g) = f x 2 ∆ j g · x = x g is clopen for every g 2 G.) By Matte-Bon- Tsankov [16], if X ⊆ SubG is any URS of G, then there exists a minimal action G y ∆ such that the stabilizer map f : ∆ ! SubG is continuous and f(∆) = X. -
On Non-Solvable Camina Pairs ∗ Zvi Arad A, Avinoam Mann B, Mikhail Muzychuk A, , Cristian Pech C
CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Journal of Algebra 322 (2009) 2286–2296 Contents lists available at ScienceDirect Journal of Algebra www.elsevier.com/locate/jalgebra On non-solvable Camina pairs ∗ Zvi Arad a, Avinoam Mann b, Mikhail Muzychuk a, , Cristian Pech c a Department of Computer Sciences and Mathematics, Netanya Academic College, University St. 1, 42365, Netanya, Israel b Einstein Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel c Department of Mathematics, Ben-Gurion University, Beer-Sheva, Israel article info abstract Article history: In this paper we study non-solvable and non-Frobenius Camina Received 27 September 2008 pairs (G, N). It is known [D. Chillag, A. Mann, C. Scoppola, Availableonline29July2009 Generalized Frobenius groups II, Israel J. Math. 62 (1988) 269–282] Communicated by Martin Liebeck that in this case N is a p-group. Our first result (Theorem 1.3) shows that the solvable residual of G/O (G) is isomorphic either Keywords: p e = Camina pair to SL(2, p ), p is a prime or to SL(2, 5), SL(2, 13) with p 3, or to SL(2, 5) with p 7. Our second result provides an example of a non-solvable and non- 5 ∼ Frobenius Camina pair (G, N) with |Op (G)|=5 and G/Op (G) = SL(2, 5).NotethatG has a character which is zero everywhere except on two conjugacy classes. Groups of this type were studies by S.M. Gagola [S.M. Gagola, Characters vanishing on all but two conjugacy classes, Pacific J. -
Arxiv:1612.04280V2 [Math.FA] 11 Jan 2018 Oal Opc Group Compact Important Locally a Is Groups, Discrete for Especially Amenability, SB878)
WEAK AMENABILITY OF LIE GROUPS MADE DISCRETE SØREN KNUDBY Abstract. We completely characterize connected Lie groups all of whose countable subgroups are weakly amenable. We also provide a characteriza- tion of connected semisimple Lie groups that are weakly amenable. Finally, we show that a connected Lie group is weakly amenable if the group is weakly amenable as a discrete group. 1. Statement of the results Weak amenability for locally compact groups was introduced by Cowling and Haagerup in [11]. The property has proven useful as a tool in operator algebras going back to Haagerup’s result on the free groups [17], results on lattices on simple Lie groups and their group von Neumann algebras [11, 18], and more recently in several results on Cartan rigidity in the theory of von Neumann algebras (see e.g. [31, 32]). Due to its many applications in operator algebras, the study of weak amenability, especially for discrete groups, is important. A locally compact group G is weakly amenable if the constant function 1 on G can be approximated uniformly on compact subsets by compactly supported Herz-Schur multipliers, uniformly bounded in norm (see Section 2 for details). The optimal uniform norm bound is the Cowling–Haagerup constant (or the weak amenability constant), denoted here Λ(G). By now, weak amenability is quite well studied, especially in the setting of connected Lie groups. The combined work of [9, 11, 12, 15, 16, 18, 19] characterizes weak amenability for simple Lie groups. For partial results in the non-simple case, we refer to [10, 25]. We record the simple case here. -
Rigidity of Some Abelian-By-Cyclic Solvable Group N Actions on T
RIGIDITY OF SOME ABELIAN-BY-CYCLIC SOLVABLE GROUP N ACTIONS ON T AMIE WILKINSON, JINXIN XUE Abstract. In this paper, we study a natural class of groups that act as affine trans- N formations of T . We investigate whether these solvable, \abelian-by-cyclic," groups N can act smoothly and nonaffinely on T while remaining homotopic to the affine actions. In the affine actions, elliptic and hyperbolic dynamics coexist, forcing a pri- ori complicated dynamics in nonaffine perturbations. We first show, using the KAM method, that any small and sufficiently smooth perturbation of such an affine action can be conjugated smoothly to an affine action, provided certain Diophantine condi- tions on the action are met. In dimension two, under natural dynamical hypotheses, we get a complete classification of such actions; namely, any such group action by Cr diffeomorphisms can be conjugated to the affine action by Cr−" conjugacy. Next, we show that in any dimension, C1 small perturbations can be conjugated to an affine action via C1+" conjugacy. The method is a generalization of the Herman theory for circle diffeomorphisms to higher dimensions in the presence of a foliation structure provided by the hyperbolic dynamics. Contents 1. Introduction 2 1.1. The affine actions of ΓB;K¯ 4 1.2. Local rigidity of ΓB;K¯ actions 6 1.3. Global rigidity 8 2. Local rigidity: proofs 14 3. The existence of the common conjugacy 18 4. Preliminaries: elliptic and hyperbolic dynamics 20 4.1. Elliptic dynamics: the framework of Herman-Yoccoz-Katznelson-Ornstein 20 4.2. Hyperbolic dynamics: invariant foliations of Anosov diffeomorphisms 24 5. -
Solvable and Nilpotent Groups
SOLVABLE AND NILPOTENT GROUPS If A; B ⊆ G are subgroups of a group G, define [A; B] to be the subgroup of G generated by all commutators f[a; b] = aba−1b−1 j a 2 A; b 2 Bg. Thus, the elements of [A; B] are finite products of such commutators and their inverses. Since [a; b]−1 = [b; a], we have [A; B] = [B; A]. If both A and B are normal subgroups of G then [A; B] is also a normal subgroup. (Clearly, c[a; b]c−1 = [cac−1; cbc−1].) Recall that a characteristic [resp fully invariant] subgroup of G means a subgroup that maps to itself under all automorphisms [resp. all endomorphisms] of G. It is then obvious that if A; B are characteristic [resp. fully invariant] subgroups of G then so is [A; B]. Define a series of normal subgroups G = G(0) ⊇ G(1) ⊇ G(2) ⊇ · · · G(0) = G; G(n+1) = [G(n);G(n)]: Thus G(n)=G(n+1) is the derived group of G(n), the universal abelian quotient of G(n). The above series of subgroups of G is called the derived series of G. Define another series of normal subgroups of G G = G0 ⊇ G1 ⊇ G2 ⊇ · · · G0 = G; Gn+1 = [G; Gn]: This second series is called the lower central series of G. Clearly, both the G(n) and the Gn are fully invariant subgroups of G. DEFINITION 1: Group G is solvable if G(n) = f1g for some n. DEFINITION 2: Group G is nilpotent if Gn = f1g for some n. -
Some Results from Group Theory
Some Results from Group Theory A.l Solvable Groups Definition A.1.1. A composition series for a group G is a sequence of sub groups G = Go2Gi 2...GA: = {1} with G/ a normal subgroup of G/_i for each /. A group G is solvable if it has a composition series with each quotient Gi/Gi-i abehan. o Example A.1.2. (1) Every abeUan group G is solvable (as we may choose Go = GandGi = {1}). (2) If G is the group of Lemma 4.6.1, G = (a, r | a^ = 1, r^~^ = 1, rar"^ = <j^} where r is a primitive root mod/?, then G is solvable: G = GODGIDG2 = {1} with Gi the cyclic subgroup generated by a. Then Gi is a normal subgroup of G with G/Gi cyclic of order p — 1 and Gi cyclic of order p. (3) If G is the group of Lemma 4.6.3, G = (a, r | a"^ = 1, r^ = 1, rorr"^ = a^) , then G is solvable: G = Go D Gi D G2 = {1} with Gi the cyclic subgroup generated by a. Then Gi is a normal subgroup of G with G/Gi cyclic of order 2 and Gi cyclic of order 4. (4) If G is a /?-group, then G is solvable. This is Lenmia A.2.4 below. (5) If G = Sn, the synmietric group on {1,..., n}, then G is solvable for n < 4. ^1 is trivial. S2 is abelian. ^3 has the composition series ^3 D A3 D {1}. 54 has the composition series ^4 D A4 D V4 D {1} where V4 = {1, (12)(34), (13)(24), (14)(23)}. -
Applications of Lie Ring Methods to Group Theory
Applications of Lie ring methods to group theory Pavel Shumyatsky1 Department of Mathematics University of Brasilia 70910-900 Brasilia - DF, Brazil e-mail: [email protected] Abstract. This article was published in 2000. Its aim is to illustrate how a Lie-theoretic result of Zelmanov enables one to treat various problems in group theory. 1. Introduction Lie rings were associated with p-groups in the 30s in the context of the Restricted Burnside Problem and since then Lie ring methods proved to be an important and very effective tool of group theory. In the last 10 years the sphere of use of Lie rings was amplified considerably, mainly due to Zelmanov’s outstanding contribution. It has been known for some time that the following assertions are equiva- lent. 1.1. Let m and n be positive integers. Then there exists a number B(m, n) depending only on m and n such that the order of any m-generator finite group of exponent n is at most B(m, n). arXiv:1706.07963v1 [math.GR] 24 Jun 2017 1.2. The class of locally nilpotent groups of exponent n is a variety. 1.3. The class of locally finite groups of exponent n is a variety. 1.4. Any residually finite group of exponent n is locally finite. 1Research supported by FAPDF and CNPq 1 The Restricted Burnside Problem is exactly the question whether the first of the above assertions is true. In 1956 P. Hall and G. Higman reduced the problem to the case of prime-power exponent [11].