On Non-Solvable Camina Pairs ∗ Zvi Arad A, Avinoam Mann B, Mikhail Muzychuk A, , Cristian Pech C
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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. -
A Class of Frobenius Groups
A CLASS OF FROBENIUS GROUPS DANIEL GORKNSTEIN 1. Introduction. If a group contains two subgroups A and B such that every element of the group is either in A or can be represented uniquely in the form aba', a, a' in A, b 5* 1 in B, we shall call the group an independent ABA-group. In this paper we shall investigate the structure of independent ABA -groups of finite order. A simple example of such a group is the group G of one-dimensional affine transformations over a finite field K. In fact, if we denote by a the transforma tion x' = cox, where co is a primitive element of K, and by b the transformation x' = —x + 1, it is easy to see that G is an independent ABA -group with respect to the cyclic subgroups A, B generated by a and b respectively. Since G admits a faithful representation on m letters {m = number of elements in K) as a transitive permutation group in which no permutation other than the identity leaves two letters fixed, and in which there is at least one permutation leaving exactly one letter fixed, G is an example of a Frobenius group. In Theorem I we shall show that this property is characteristic of independent ABA-groups. In a Frobenius group on m letters, the set of elements whose order divides m forms a normal subgroup, called the regular subgroup. In our example, the regular subgroup M of G consists of the set of translations, and hence is an Abelian group of order m = pn and of type (p, p, . -
A Note on Frobenius Groups
Journal of Algebra 228, 367᎐376Ž. 2000 doi:10.1006rjabr.2000.8269, available online at http:rrwww.idealibrary.com on A Note on Frobenius Groups Paul Flavell The School of Mathematics and Statistics, The Uni¨ersity of Birmingham, CORE Birmingham B15 2TT, United Kingdom Metadata, citation and similar papers at core.ac.uk E-mail: [email protected] Provided by Elsevier - Publisher Connector Communicated by George Glauberman Received January 25, 1999 1. INTRODUCTION Two celebrated applications of the character theory of finite groups are Burnside's p␣q -Theorem and the theorem of Frobenius on the groups that bear his name. The elegant proofs of these theorems were obtained at the beginning of this century. It was then a challenge to find character-free proofs of these results. This was achieved for the p␣q -Theorem by Benderwx 1 , Goldschmidt wx 3 , and Matsuyama wx 8 in the early 1970s. Their proofs used ideas developed by Feit and Thompson in their proof of the Odd Order Theorem. There is no known character-free proof of Frobe- nius' Theorem. Recently, CorradiÂÂ and Horvathwx 2 have obtained some partial results in this direction. The purpose of this note is to prove some stronger results. We hope that this will stimulate more interest in this problem. Throughout the remainder of this note, we let G be a finite Frobenius group. That is, G contains a subgroup H such that 1 / H / G and H l H g s 1 for all g g G y H. A subgroup with these properties is called a Frobenius complement of G. -
Lecture 14. Frobenius Groups (II)
Quick Review Proof of Frobenius’ Theorem Heisenberg groups Lecture 14. Frobenius Groups (II) Daniel Bump May 28, 2020 Quick Review Proof of Frobenius’ Theorem Heisenberg groups Review: Frobenius Groups Definition A Frobenius Group is a group G with a faithful transitive action on a set X such that no element fixes more than one point. An action of G on a set X gives a homomorphism from G to the group of bijections X (the symmetric group SjXj). In this definition faithful means this homomorphism is injective. Let H be the stabilizer of a point x0 2 X. The group H is called the Frobenius complement. Today we will prove: Theorem (Frobenius (1901)) A Frobenius group G is a semidirect product. That is, there exists a normal subgroup K such that G = HK and H \ K = f1g. Quick Review Proof of Frobenius’ Theorem Heisenberg groups Review: The mystery of Frobenius’ Theorem Since Frobenius’ theorem doesn’t require group representation theory in its formulation, it is remarkable that no proof has ever been found that doesn’t use representation theory! Web links: Frobenius groups (Wikipedia) Fourier Analytic Proof of Frobenius’ Theorem (Terence Tao) Math Overflow page on Frobenius’ theorem Frobenius Groups (I) (Lecture 14) Quick Review Proof of Frobenius’ Theorem Heisenberg groups The precise statement Last week we introduced the notion of a Frobenius group. This is a group G that acts transitively on a set X in which no element except the identity fixes more than one point. Let H be the isotropy subgroup of an element, and let be K∗ [ f1g where K∗ is the set of elements with no fixed points. -
12.6 Further Topics on Simple Groups 387 12.6 Further Topics on Simple Groups
12.6 Further Topics on Simple groups 387 12.6 Further Topics on Simple Groups This Web Section has three parts (a), (b) and (c). Part (a) gives a brief descriptions of the 56 (isomorphism classes of) simple groups of order less than 106, part (b) provides a second proof of the simplicity of the linear groups Ln(q), and part (c) discusses an ingenious method for constructing a version of the Steiner system S(5, 6, 12) from which several versions of S(4, 5, 11), the system for M11, can be computed. 12.6(a) Simple Groups of Order less than 106 The table below and the notes on the following five pages lists the basic facts concerning the non-Abelian simple groups of order less than 106. Further details are given in the Atlas (1985), note that some of the most interesting and important groups, for example the Mathieu group M24, have orders in excess of 108 and in many cases considerably more. Simple Order Prime Schur Outer Min Simple Order Prime Schur Outer Min group factor multi. auto. simple or group factor multi. auto. simple or count group group N-group count group group N-group ? A5 60 4 C2 C2 m-s L2(73) 194472 7 C2 C2 m-s ? 2 A6 360 6 C6 C2 N-g L2(79) 246480 8 C2 C2 N-g A7 2520 7 C6 C2 N-g L2(64) 262080 11 hei C6 N-g ? A8 20160 10 C2 C2 - L2(81) 265680 10 C2 C2 × C4 N-g A9 181440 12 C2 C2 - L2(83) 285852 6 C2 C2 m-s ? L2(4) 60 4 C2 C2 m-s L2(89) 352440 8 C2 C2 N-g ? L2(5) 60 4 C2 C2 m-s L2(97) 456288 9 C2 C2 m-s ? L2(7) 168 5 C2 C2 m-s L2(101) 515100 7 C2 C2 N-g ? 2 L2(9) 360 6 C6 C2 N-g L2(103) 546312 7 C2 C2 m-s L2(8) 504 6 C2 C3 m-s -
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. -
Quasi P Or Not Quasi P? That Is the Question
Rose-Hulman Undergraduate Mathematics Journal Volume 3 Issue 2 Article 2 Quasi p or not Quasi p? That is the Question Ben Harwood Northern Kentucky University, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Harwood, Ben (2002) "Quasi p or not Quasi p? That is the Question," Rose-Hulman Undergraduate Mathematics Journal: Vol. 3 : Iss. 2 , Article 2. Available at: https://scholar.rose-hulman.edu/rhumj/vol3/iss2/2 Quasi p- or not quasi p-? That is the Question.* By Ben Harwood Department of Mathematics and Computer Science Northern Kentucky University Highland Heights, KY 41099 e-mail: [email protected] Section Zero: Introduction The question might not be as profound as Shakespeare’s, but nevertheless, it is interesting. Because few people seem to be aware of quasi p-groups, we will begin with a bit of history and a definition; and then we will determine for each group of order less than 24 (and a few others) whether the group is a quasi p-group for some prime p or not. This paper is a prequel to [Hwd]. In [Hwd] we prove that (Z3 £Z3)oZ2 and Z5 o Z4 are quasi 2-groups. Those proofs now form a portion of Proposition (12.1) It should also be noted that [Hwd] may also be found in this journal. Section One: Why should we be interested in quasi p-groups? In a 1957 paper titled Coverings of algebraic curves [Abh2], Abhyankar conjectured that the algebraic fundamental group of the affine line over an algebraically closed field k of prime characteristic p is the set of quasi p-groups, where by the algebraic fundamental group of the affine line he meant the family of all Galois groups Gal(L=k(X)) as L varies over all finite normal extensions of k(X) the function field of the affine line such that no point of the line is ramified in L, and where by a quasi p-group he meant a finite group that is generated by all of its p-Sylow subgroups. -
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)}.