The Arithmetic of Fundamental Groups
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Math 602, Fall 2002. BIII (A). We First Prove the Following Proposition
Homework I (due September 30), Math 602, Fall 2002. BIII (a). We first prove the following proposition: Proposition 1.1 Given a group G, for any finite normal subgroup, H, of G and any p-Sylow subgroup P of H, we have G = NG(P )H. Proof . For every g ∈ G, clearly, g−1P g is a subgroup of g−1Hg = H, since H is normal in G. Since |g−1P g| = |P |, the subgroup g−1P g is also a p-Sylow subgroup of H. By Sylow II, g−1P g is some conjugate of P in H, i.e., g−1P g = hP h−1 for some h ∈ H. −1 −1 Thus, ghP h g = P , which implies that gh ∈ NG(P ), so g ∈ NG(P )H. Since the reasoning holds for every g ∈ G, we get G = NG(P )H. Now, since H is normal in G, by the second homomorphism theorem, we know that ∼ G/H = (NG(P )H)/H = NG(P )/(NG(P ) ∩ H). Moreover, it is clear that NG(P ) ∩ H = ∼ NH (P ), so G/H = NG(P )/NH (P ), as desired. (b) We shall prove that every p-Sylow subgroup of Φ(G) is normal in Φ(G) (and in fact, in G). From this, we will deduce that Φ(G) has property (N). Indeed, if we inspect the proof of the proposition proved in class stating that if G is a finite group that has (N), then G is isomorphic to the product of its p-Sylow subgroups, we see that this proof only depends on the fact that every p-Sylow subgroup of G is normal in G. -
A CHARACTERISTIC SUBGROUP of a P-GROUP
Pacific Journal of Mathematics A CHARACTERISTIC SUBGROUP OF A p-GROUP CHARLES RAY HOBBY Vol. 10, No. 3 November 1960 A CHARACTERISTIC SUBGROUP OF A p-GROVP CHARLES HOBBY If x, y are elements and H, K subsets of the p-group G, we shall denote by [x, y] the element y~px~p(xy)p of G, and by [H, K] the sub- group of G generated by the set of all [h, k] for h in H and k in K. We call a p-group G p-abelίan if (xy)p = xpyp for all elements x, y of G. If we let Θ(G) — [G, G] then #(G) is a characteristic subgroup of G and Gjθ{G) is p-abelian. In fact, Θ(G) is the minimal normal subgroup N of G for which G/AΓ is p-abelian. It is clear that Θ(G) is contained in the derived group of G, and G/Θ(G) is regular in the sense of P. Hall [3] Theorem 1 lists some elementary properties of p-abelian groups. These properties are used to obtain a characterization of p-groups G (for p > 3) in which the subgroup generated by the pth powers of elements of G coincides with the Frattini subgroup of G (Theorems 2 and 3). A group G is said to be metacyclic if there exists a cyclic normal sub- group N with G/N cyclic. Theorem 4 states that a p-group G, for p > 2, is metacyclic if and only if Gjθ(G) is metacyclic. -
Automorphism Groups of Free Groups, Surface Groups and Free Abelian Groups
Automorphism groups of free groups, surface groups and free abelian groups Martin R. Bridson and Karen Vogtmann The group of 2 × 2 matrices with integer entries and determinant ±1 can be identified either with the group of outer automorphisms of a rank two free group or with the group of isotopy classes of homeomorphisms of a 2-dimensional torus. Thus this group is the beginning of three natural sequences of groups, namely the general linear groups GL(n, Z), the groups Out(Fn) of outer automorphisms of free groups of rank n ≥ 2, and the map- ± ping class groups Mod (Sg) of orientable surfaces of genus g ≥ 1. Much of the work on mapping class groups and automorphisms of free groups is motivated by the idea that these sequences of groups are strongly analogous, and should have many properties in common. This program is occasionally derailed by uncooperative facts but has in general proved to be a success- ful strategy, leading to fundamental discoveries about the structure of these groups. In this article we will highlight a few of the most striking similar- ities and differences between these series of groups and present some open problems motivated by this philosophy. ± Similarities among the groups Out(Fn), GL(n, Z) and Mod (Sg) begin with the fact that these are the outer automorphism groups of the most prim- itive types of torsion-free discrete groups, namely free groups, free abelian groups and the fundamental groups of closed orientable surfaces π1Sg. In the ± case of Out(Fn) and GL(n, Z) this is obvious, in the case of Mod (Sg) it is a classical theorem of Nielsen. -
Free and Linear Representations of Outer Automorphism Groups of Free Groups
Free and linear representations of outer automorphism groups of free groups Dawid Kielak Magdalen College University of Oxford A thesis submitted for the degree of Doctor of Philosophy Trinity 2012 This thesis is dedicated to Magda Acknowledgements First and foremost the author wishes to thank his supervisor, Martin R. Bridson. The author also wishes to thank the following people: his family, for their constant support; David Craven, Cornelia Drutu, Marc Lackenby, for many a helpful conversation; his office mates. Abstract For various values of n and m we investigate homomorphisms Out(Fn) ! Out(Fm) and Out(Fn) ! GLm(K); i.e. the free and linear representations of Out(Fn) respectively. By means of a series of arguments revolving around the representation theory of finite symmetric subgroups of Out(Fn) we prove that each ho- momorphism Out(Fn) ! GLm(K) factors through the natural map ∼ πn : Out(Fn) ! GL(H1(Fn; Z)) = GLn(Z) whenever n = 3; m < 7 and char(K) 62 f2; 3g, and whenever n + 1 n > 5; m < 2 and char(K) 62 f2; 3; : : : ; n + 1g: We also construct a new infinite family of linear representations of Out(Fn) (where n > 2), which do not factor through πn. When n is odd these have the smallest dimension among all known representations of Out(Fn) with this property. Using the above results we establish that the image of every homomor- phism Out(Fn) ! Out(Fm) is finite whenever n = 3 and n < m < 6, and n of cardinality at most 2 whenever n > 5 and n < m < 2 . -
Workshop on Rational Points and Brauer-Manin Obstruction
WORKSHOP ON RATIONAL POINTS AND BRAUER-MANIN OBSTRUCTION OBSTRUCTION GUYS ABSTRACT. Here we present an extended version of the notes taken by the seminars organized during the winter semester of 2015. The main goal is to provide a quick introduction to the theory of Brauer-Manin Obstructions (following the book of Skorobogatov and some more recent works). We thank Professor Harari for his support and his active participation in this exciting workshop. Contents 1. Talk 1: A first glance, Professor Harari 3 1.1. Galois and Étale Cohomology............................... 3 1.1.1. Group Cohomology ................................ 3 1.1.2. Modified/Tate Group................................ 3 1.1.3. Restriction-inflation ................................ 3 1.1.4. Profinite Groups .................................. 4 1.1.5. Galois Case..................................... 4 1.1.6. Étale Cohomology................................. 4 1.1.7. Non abelian Cohomology ............................. 4 1.2. Picard Group, Brauer Group................................ 5 1.3. Obstructions ........................................ 6 1.4. Miscellaneous ....................................... 7 1.4.1. Weil Restrictions.................................. 7 1.4.2. Induced Module are acyclic ............................ 7 2. Talk 2: Torsors, Emiliano Ambrosi 8 2.1. Torsors........................................... 8 2.2. Torsors and cohomology.................................. 10 2.3. Torsors and rational points................................. 11 3. Talk 3: Descent -
P-Groups with a Unique Proper Non-Trivial Characteristic Subgroup ∗ S.P
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Algebra 348 (2011) 85–109 Contents lists available at SciVerse ScienceDirect Journal of Algebra www.elsevier.com/locate/jalgebra p-Groups with a unique proper non-trivial characteristic subgroup ∗ S.P. Glasby a,P.P.Pálfyb, Csaba Schneider c, a Department of Mathematics, Central Washington University, WA 98926-7424, USA b Alfréd Rényi Institute of Mathematics, 1364 Budapest, Pf. 127, Hungary c Centro de Álgebra da Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal article info abstract Article history: We consider the structure of finite p-groups G having precisely Received 23 July 2010 three characteristic subgroups, namely 1, Φ(G) and G.Thestruc- Available online 21 October 2011 ture of G varies markedly depending on whether G has exponent Communicated by Aner Shalev p or p2, and, in both cases, the study of such groups raises deep problems in representation theory. We present classification the- MSC: 20D15 orems for 3- and 4-generator groups, and we also study the ex- 2 20C20 istence of such r-generator groups with exponent p for various 20E15 values of r. The automorphism group induced on the Frattini quo- 20F28 tient is, in various cases, related to a maximal linear group in Aschbacher’s classification scheme. Keywords: © 2011 Elsevier Inc. All rights reserved. UCS-groups Finite p-groups Automorphism groups Characteristic subgroups 1. Introduction Taunt [Tau55] considered groups having precisely three characteristic subgroups. As such groups have a unique proper non-trivial characteristic subgroup, he called these UCS-groups. -
The Brauer-Manin Obstruction of Del Pezzo Surfaces of Degree 4
The Brauer-Manin Obstruction of del Pezzo Surfaces of Degree 4 Manar Riman August 26, 2017 Abstract Let k be a global field, and X a del Pezzo surface of degree 4. Under the assumption of existence of a proper regular model, we prove in particular cases that if the Brauer-Manin set is empty over k then it is empty over L where L is an odd degree extension of k. This problem provides evidence for a conjecture of Colliot-Th´el`eneand Sansuc about the sufficiency of the Brauer-Manin obstruction to existence of rational points on a rational surface. Contents 1 Introduction 2 2 Rational Points 3 2.1 Brauer Group of a Field . .3 2.1.1 Cyclic Algebras . .5 2.1.2 The Brauer Group of a Local Field . .6 2.1.3 The Brauer Group of a Global Field . .6 2.2 The Brauer Group of a Scheme . .6 2.2.1 Hochschild-Serre Spectral Sequence . .7 2.2.2 Residue Maps . .8 2.2.3 The Brauer-Manin Obstruction . .9 2.3 Torsors . .9 3 Del Pezzo Surfaces 10 3.1 Geometry . 10 3.1.1 Picard Group . 13 3.2 Arithmetic . 14 4 The Brauer-Manin Set of a Del Pezzo surface of degree 4 14 4.1 The Main Problem . 14 4.2 Geometry and the Evaluation Map . 16 4.2.1 Purity: Gysin Sequence . 16 4.2.2 Constant Evaluation . 16 1 1 Introduction Let X be a smooth, projective, geometrically integral variety over a global field k. We define the 1 ad`elering to be the restricted product pk ; q where Ω is the set of places of k. -
Finite Descent Obstructions and Rational Points on Curves Draft Version No
FINITE DESCENT OBSTRUCTIONS AND RATIONAL POINTS ON CURVES DRAFT VERSION NO. 8 MICHAEL STOLL Abstract. Let k be a number field and X a smooth projective k-variety. In this paper, we discuss the information obtainable from descent via torsors under finite k-group schemes on the location of the k-rational points on X within the adelic points. We relate finite descent obstructions to the Brauer-Manin obstruction; in particular, we prove that on curves, the Brauer set equals the set cut out by finite abelian descent. We show that on many curves of genus ≥ 2, the information coming from finite abelian descent cuts out precisely the rational points. If such a curve fails to have rational points, this is then explained by the Brauer-Manin obstruction. In the final section of this paper, we present some conjectures that are motivated by our results. Contents 1. Introduction 2 2. Preliminaries 4 3. Some results on abelian varieties 6 4. Torsors and twists 12 5. Finite descent conditions 14 6. Finite descent conditions and rational points 22 7. Relation with the Brauer-Manin obstruction 27 8. Finite descent conditions on curves 37 9. Some conjectures 39 References 51 Date: November 22, 2006. 2000 Mathematics Subject Classification. Primary 11G10, 11G30, 11G35, 14G05, 14G25, 14H30; Secondary 11R34, 14H25, 14J20, 14K15. Key words and phrases. rational points, descent obstructions, coverings, twists, torsors under finite group schemes, Brauer-Manin obstruction, birational section conjecture. 1 FINITE DESCENT AND RATIONAL POINTS ON CURVES 2 1. Introduction In this paper we explore what can be deduced about the set of rational points on a curve (or a more general variety) from a knowledge of its finite ´etalecoverings. -
7. Minimal Normal Subgroups and the Socle Throughout This Section G Is A
34 NICK GILL 7. Minimal normal subgroups and the socle Throughout this sectionG is a nontrivial group. A minimal normal subgroup ofG is a normal subgroup K= 1 ofG which does not contain any other nontrivial normal subgroup ofG. The socle, soc(G) is the subgroup� generated by the set of all minimal normal subgroups (ifG has no minimal normal subgroups, then we set soc(G) = 1). (E7.1)Find the socle ofD 10,A 4,S 4,S 4 Z. × (E7.2)Give an example of a group that has no minimal normal subgroup. (E7.3)IfG is the direct product of a (possibly infinite) number of finite simple groups, then what is soc(G)? (E7.4)Give a characteri ation of almost simple groups in terms of their socle. Clearly soc(G) is a characteristic subgroup ofG. 26 Theorem 7.1. LetG be a nontrivial finite group. (1) IfK is a minimal normal subgroup ofG, andL is any normal subgroup ofG, then eitherK L or K,L =K L. 27 ≤ (2) There� exist� minimal× normal subgroupsK ,...,K ofG such that soc(G)=K K . 1 m 1 × · · · m (3) Every minimal normal subgroupK ofG is a direct productK=T 1 T k where theT i are simple normal subgroups ofK which are conjugate inG. × · · · (4) If the subgroupsK i in (2) are all nonabelian, thenK 1,...,Km are the only minimal normal sub- groups ofG. Proof. (1) #inceK L�G, the minimality ofK implies that eitherK L orK L= 1 . $n the latter case, since∩ K andL are both normal, we have that K,L =≤KL=K∩L. -
Algebraic Families of Nonzero Elements of Shafarevich-Tate Groups
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000{000 S 0894-0347(XX)0000-0 ALGEBRAIC FAMILIES OF NONZERO ELEMENTS OF SHAFAREVICH-TATE GROUPS JEAN-LOUIS COLLIOT-THEL´ ENE` AND BJORN POONEN 1. Introduction Around 1940, Lind [Lin] and (independently, but shortly later) Reichardt [Re] discovered that some genus 1 curves over Q, such as 2y2 = 1 − 17x4; violate the Hasse principle; i.e., there exist x; y 2 R satisfying the equation, and for each prime p there exist x; y 2 Qp satisfying the equation, but there do not exist x; y 2 Q satisfying the equation. In fact, even the projective nonsingular model has no rational points. We address the question of constructing algebraic families of such examples. Can one find an equation in x and y whose coefficients are rational functions of a parameter t, such that specializing t to any rational number results in a genus 1 curve violating the Hasse principle? The answer is yes, even if we impose a non-triviality condition; this is the content of our Theorem 1.2. Any genus 1 curve X is a torsor for its Jacobian E. Here E is an elliptic curve: a genus 1 curve with a rational point. If X violates the Hasse principle, then X moreover represents a nonzero element of the Shafarevich-Tate group X(E). We may relax our conditions by asking for families of torsors of abelian varieties. In this case we can find a family with stronger properties: one in which specializing the parameter to any number α 2 Q of odd degree over Q results in a torsor of an abelian variety over Q(α) violating the Hasse principle. -
The Brauer-Manin Obstruction and the Fibration Method - Lecture by Jean-Louis Colliot-Thel´ Ene`
THE BRAUER-MANIN OBSTRUCTION AND THE FIBRATION METHOD - LECTURE BY JEAN-LOUIS COLLIOT-THEL´ ENE` These are informal notes on the lecture I gave at IU Bremen on July 14th, 2005. Steve Donnelly prepared a preliminary set of notes, which I completed. 1. The Brauer-Manin obstruction 2 For a scheme X, the Brauer group Br X is H´et(X, Gm) ([9]). When X is a regular, noetherian, separated scheme, this coincides with the Azumaya Brauer group. If F ia a field, then Br(Spec F ) = Br F = H2(Gal(F /F ),F ×). Let k be a field and X a k-variety. For any field F containing k, each element A ∈ Br X gives rise to an “evaluation map” evA : X(F ) → Br F . For a number field k, class field theory gives a fundamental exact sequence L invv . 0 −−−→ Br k −−−→ all v Br kv −−−→ Q/Z −−−→ 0 That the composite map from Br k to Q/Z is zero is a generalization of the quadratic reciprocity law. Now suppose X is a projective variety defined over a number field k, and A ∈ Br X. One has the commutative diagram Q X(k) −−−→ v X(kv) evA evA y y Br k −−−→ ⊕v Br kv −−−→ Q/Z The second vertical map makes sense because, if X is projective, then for each fixed A, evA : X(kv) → Br kv is the zero map for all but finitely many v. Q Let ΘA : v X(kv) → Q/Z denote the composed map. Thus X(k) ⊆ ker ΘA for all A. -
Operations on Maps, and Outer Automorphisms
JOURNAL OF COMBINATORIAL THEORY, Series B 35, 93-103 (1983) Operations on Maps, and Outer Automorphisms G. A. JONES AND J. S. THORNTON Department of Mathematics, University of Southampton, Southampton SO9 SNH, England Communicated by the Editors Received December 20, 1982 By representing maps on surfaces as transitive permutation representations of a certain group r, it is shown that there are exactly six invertible operations (such as duality) on maps; they are induced by the outer automorphisms of r, and form a group isomorphic to S, Various consequences are deduced, such as the result that each finite map has a finite reflexible cover which is invariant under all six operations. 1. INTRODUCTION There is a well-known duality for maps on surfaces, interchanging vertices and faces, so that a map ,& of type (m, n) is transformed into its dual map ,H* of type (n, m) while retaining certain important features such as its automorphism group. Recently topological descriptions have been given for other similar operations on maps, firstly by Wilson [ 1 I] for regular and reflexible maps, and later by Lins [7] for all maps; they each exhibit four more invertible operations which, together with the above duality and the identity operation, form a group isomorphic to S,. The object of this note is to show that these operations arise naturally in algebraic map theory: maps may be regarded as transitive permutation representations of a certain group r, and the outer automorphism group Out(T) = Aut(T)/Inn(T) g S, permutes these representations, inducing the six operations on maps.