DOCSLIB.ORG

On the Local Theory of Profinite Groups

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On the Local Theory of Profinite Groups Western Michigan University ScholarWorks at WMU Dissertations Graduate College 12-2020 On the Local Theory of Profinite Groups Mohammad Shatnawi Western Michigan University, [email protected] Follow this and additional works at: https://scholarworks.wmich.edu/dissertations Part of the Algebra Commons, and the Number Theory Commons Recommended Citation Shatnawi, Mohammad, "On the Local Theory of Profinite Groups" (2020). Dissertations. 3691. https://scholarworks.wmich.edu/dissertations/3691 This Dissertation-Open Access is brought to you for free and open access by the Graduate College at ScholarWorks at WMU. It has been accepted for inclusion in Dissertations by an authorized administrator of ScholarWorks at WMU. For more information, please contact [email protected]. ON THE LOCAL THEORY OF PROFINITE GROUPS by Mohammad Shatnawi A dissertation submitted to the Graduate College in partial fulfilment of the requirements for the degree of doctor of Philosophy Mathematics Western Michigan University December 2020 Doctoral Committee: Clifton Edgar Ealy Jr, PhD., Chair John Martino, PhD. Jeffry Strom, PhD. Justin Lynd, PhD. ON THE LOCAL THEORY OF PROFINITE GROUPS Mohammad Shatnawi, Ph.D. Western Michigan University, 2020 Let G be a finite group, and H be a subgroup of G. The transfer homomorphism emerges from the natural action of G on the cosets of H. The transfer was first introduced by Schur in 1902 [22] as a construction in group theory, which produce a homomorphism from a finite group G into H/H0 an abelian group where H is a subgroup of G and H0 is the derived group of H. One important first application is Burnside’s normal p- complement theorem [5] in 1911, although he did not use the transfer homomorphism explicitly to prove it. Burnside Theorem. Let G be a finite group, and let P be a sylow p-subgroup that is contained in the center of its normalizer, then G has a normal subgroup H which has elements of P as its coset representatives. Emil Artin in 1929 [1] Extended the definition to the situation G is infinite and H is a subgroup of G of finite index. The first place the transfer homomorphism appeared in a textbook was in 1937, Lehrbuch der Gruppentheorie, by Hans Zassenhaus [28]. In 1959 the transfer was popularized in the mathematical community in America by Marshall Hall in the textbook The Theory of Groups [14] where the transfer ”in German Die Verlagerung” as a homomorphism from a group G into a subgroup H of finite index was introduced. In an effort to define the transfer homomorphism for profinite groups, Oliver Schirokauer in 1996 [21] published a paper in which he presented a new definition for the standard cohomological transfer as an integral. In this study we will give a definition of the transfer homomorphism for profinite groups which is an analog of Marshall Hall’s definition of the transfer homomorphism for finite groups. Our definition depends on the group action and structure, in relation to axiom of choice and ordinal numbers, using the permanent map. © 2020 Mohammad Shatnawi ACKNOWLEDGMENTS Firstly, I would like to express my sincere gratitude to my advisor Prof. Clifton Edgar Ealy, Jr for the continuous support of my Ph.D study, for his patience, motivation, and immense knowledge. His guidance helped me in all the time of research and writing of this thesis. I could not have imagined having a better advisor and mentor for my Ph.D study. I’m very grateful to my father and mother Ali and Rifqah Shatnawi, my sisters Rayah, Nusaibah, Sana’, Areej, Azhar, Fatimah, and Bara’ah, my brothers Abraham and Abed ElRahman for their endless support and love without which I won’t be able to reach this point. I’m very grateful to have my wife Tasnim Alhaddadein, who shared my journey and supported me with all her strength to fulfill our dream. I’m grateful to have Elias, Ryan, Farah and Jawan in my life who supported me in their own way. I would like to thank the committee members for taking the time in reading my dissertation. I would like to express my gratitude to Dr. Kirsty Eisenhart, thank you for being my sounding board and help. Finally, I would like to thank the Mathematics department faculty, staff and graduate students at Western Michigan University for the unique experience and the wonderful journey which I will never forget. Mohammad Shatnawi ii TABLE OF CONTENTS ACKNOWLEDGMENTS ii 1 Statement of Main Results 1 2 Introduction 4 2.1 Finite Groups . 4 2.2 Topological Groups . 10 2.3 Profinite Groups . 12 2.4 Sylow Theory for Profinite Groups . 18 3 Pro-p Groups 22 3.1 Preliminaries . 22 3.2 Example: The Profinite Dihedral 2-Group . 23 3.3 Example: The Profinite Generalized Quaternion Group . 27 3.4 Profinite Hamiltonian Groups . 31 3.5 Sylow Intersections in Profinite Groups . 33 4 The Transfer Homomorphism for Profinite Groups 36 4.1 Necessary Background in Set Theory . 36 4.2 The Permanent Mapping . 38 4.3 The Transfer of Profinite Groups . 40 4.4 Applications of the Profinite Transfer . 48 TABLE OF CONTENTS - Continued 5 Future Work 54 Bibliography 56 Chapter 1 Statement of Main Results In Chapter 3, we give some examples of pro-2 groups like the profinite dihedral group: Example 1.0.1. The profinite dihedral group D∗ is the inverse limit of the inverse system of finite dihedral groups, D , n N. n ∈ Example 1.0.2. The inverse limit of the dihedral 2-groups D2n is isomorphic to D2∞ . and we give an extension of the generalized quaternion group as a profinite group: Example 1.0.3. The profinite generalized quaternion group Q∗ is the inverse limit of finite generalized quaternion groups. We characterize the profinite Hamiltonian groups as an inverse limit of finite Hamiltonian groups. Theorem 1.0.4. Let H be a pro-Hamiltonian group. Then H is a profinite abelian group, or H = Q E2 A where E2 is an elementary abelian pro-2 group, Q is ∗ × ∗ × ∗ ∗ ∗ the profinite generalized quaternion group and A∗ is a profinite abelian group in which every element is of odd Steintz prime order. We also extend some results of sylow intersections to profinite groups: Lemma 1.0.5. Let G be a profinite group, and let K = (S S Syl (G)). Then: | ∈ p T 1 (a)K E G, and K is a pro-p subgroup of G. (b) K contains every normal pro-p subgroup of G. Theorem 1.0.6. Let G be a profinite group, let D = P Q be a maximal sylow p- ∩ intersection of P,Q Sylp(G) with P , Q. Then: ∈ (a) D is a proper subgroup of each sylow p-subgroup of NG(D). (b) Any two distinct sylow p-subgroups of NG(D) have intersection D. (c) There exists a one-to-one correspondence between sylow p-subgroups of G con- taining D and the sylow p-subgroups of NG(D). The main result of this study is in chapter 4 in which we introduce the transfer for profinite groups using the permanent map. Let G be a profinite group, H be a closed subgroup of G of not necessarily finite index, and K be a normal subgroup of H such that H/K is abelian. Let Y be a transversal of H in G and Y = n SN, where SN is | | ∈ the set of all Steinitz numbers. For an infinite Steinitz number n we say that a monomial (n n)-matrix A over a commutative ring with identity R is known if the limit × lim ∏ am i n m i → ≤ exists and converges in R regardless of the order of the indices, where a R is the m ∈ nonzero entry in the m-th row of A. Let M (H/K) be the group of all (n n) known monomial matrices with entries n × from H/K. We define the transfer homomorphism of profinite groups as a composition of the monomial representation µ : G M (H/K) → n 2 and the permanent map perm : M (H/K) H/K. n n → This definition is an analog for the one of the transfer for finite groups in [14]. In an effort to define the transfer homomorphism for profinite groups, Oliver Schirokauer in 1996 [21] published a paper in which he presented a new definition for the stan- dard cohomological transfer as an integral, our aim is to define the transfer using the group structure and the permanent map. In [10] the authors proved Burnside Normal p-Complement Theorem for Profinite Groups Theorem 1.0.7 (Burnside Normal p-Complement Theorem for Profinite Groups). Let G be a profinite group, and let P be a sylow pro-p subgroup that is contained in the center of its normalizer, then G has a closed normal subgroup H which has elements of P as its coset representatives. without using the transfer, instead they use the following lemma Lemma 1.0.8 ([10]). Let G be an inverse limit of finite groups Gi where i is in a directed set I. Then G has a normal sylow p-complement if and only if Gi has a normal sylow p-complement for all i I. ∈ We give a proof of this theorem using the transfer following the work in [14] chapter 14, also we used the transfer to prove Theorem 1.0.9 (Focal Subgroup Theorem for Profinite Groups). Let G be a profinite group, P a pro-p sylow subgroup of G and G0 be the derived group of G. Then G V (G) P/P G0 P ∩ which was also been proved in [10] for profinite groups by extending from the finite version without using the transfer.
Load more

Recommended publications
  • 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.
    [Show full text]
  • Chapter 9 Quadratic Residues
    Chapter 9 Quadratic Residues 9.1 Introduction Definition 9.1. We say that a 2 Z is a quadratic residue mod n if there exists b 2 Z such that a ≡ b2 mod n: If there is no such b we say that a is a quadratic non-residue mod n. Example: Suppose n = 10. We can determine the quadratic residues mod n by computing b2 mod n for 0 ≤ b < n. In fact, since (−b)2 ≡ b2 mod n; we need only consider 0 ≤ b ≤ [n=2]. Thus the quadratic residues mod 10 are 0; 1; 4; 9; 6; 5; while 3; 7; 8 are quadratic non-residues mod 10. Proposition 9.1. If a; b are quadratic residues mod n then so is ab. Proof. Suppose a ≡ r2; b ≡ s2 mod p: Then ab ≡ (rs)2 mod p: 9.2 Prime moduli Proposition 9.2. Suppose p is an odd prime. Then the quadratic residues coprime to p form a subgroup of (Z=p)× of index 2. Proof. Let Q denote the set of quadratic residues in (Z=p)×. If θ :(Z=p)× ! (Z=p)× denotes the homomorphism under which r 7! r2 mod p 9–1 then ker θ = {±1g; im θ = Q: By the first isomorphism theorem of group theory, × jkerθj · j im θj = j(Z=p) j: Thus Q is a subgroup of index 2: p − 1 jQj = : 2 Corollary 9.1. Suppose p is an odd prime; and suppose a; b are coprime to p. Then 1. 1=a is a quadratic residue if and only if a is a quadratic residue.
    [Show full text]
  • 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.
    [Show full text]
  • On the Similarity Transformation Between a Matirx and Its Transpose
    Pacific Journal of Mathematics ON THE SIMILARITY TRANSFORMATION BETWEEN A MATIRX AND ITS TRANSPOSE OLGA TAUSSKY AND HANS ZASSENHAUS Vol. 9, No. 3 July 1959 ON THE SIMILARITY TRANSFORMATION BETWEEN A MATRIX AND ITS TRANSPOSE OLGA TAUSSKY AND HANS ZASSENHAUS It was observed by one of the authors that a matrix transforming a companion matrix into its transpose is symmetric. The following two questions arise: I. Does there exist for every square matrix with coefficients in a field a non-singular symmetric matrix transforming it into its transpose ? II. Under which conditions is every matrix transforming a square matrix into its transpose symmetric? The answer is provided by THEOREM 1. For every n x n matrix A — (aik) with coefficients in a field F there is a non-singular symmetric matrix transforming A into its transpose Aτ'. THEOREM 2. Every non-singular matrix transforming A into its transpose is symmetric if and only if the minimal polynomial of A is equal to its characteristic polynomial i.e. if A is similar to a com- panion matrix. Proof. Let T = (ti1c) be a solution matrix of the system Σ(A) of the linear homogeneous equations. (1) TA-ATT=O ( 2 ) T - Tτ = 0 . The system Σ(A) is equivalent to the system (3) TA-ATTT = 0 ( 4 ) T - Tτ = 0 which states that Γand TA are symmetric. This system involves n2 — n equations and hence is of rank n2 — n at most. Thus there are at least n linearly independent solutions of Σ(A).1 On the other hand it is well known that there is a non-singular matrix To satisfying - A* , Received December 18, 1958.
    [Show full text]
  • Fundamental Groups of Schemes
    Fundamental Groups of Schemes Master thesis under the supervision of Jilong Tong Lei Yang Universite Bordeaux 1 E-mail address: [email protected] Chapter 1. Introduction 3 Chapter 2. Galois categories 5 1. Galois categories 5 §1. Definition and elementary properties. 5 §2. Examples and the main theorem 7 §2.1. The topological covers 7 §2.2. The category C(Π) and the main theorem 7 2. Galois objects. 8 3. Proof of the main theorem 12 4. Functoriality of Galois categories 15 Chapter 3. Etale covers 19 1. Some results in scheme theory. 19 2. The category of étale covers of a connected scheme 20 3. Reformulation of functoriality 22 Chapter 4. Properties and examples of the étale fundamental group 25 1. Spectrum of a field 25 2. The first homotopy sequence. 25 3. More examples 30 §1. Normal base scheme 30 §2. Abelian varieties 33 §2.1. Group schemes 33 §2.2. Abelian Varieties 35 §3. Geometrically connected schemes of finite type 39 4. G.A.G.A. theorems 39 Chapter 5. Structure of geometric fundamental groups of smooth curves 41 1. Introduction 41 2. Case of characteristic zero 42 §1. The case k = C 43 §2. General case 43 3. Case of positive characteristic 44 (p0) §1. π1(X) 44 §1.1. Lifting of curves to characteristic 0 44 §1.2. the specialization theory of Grothendieck 45 §1.3. Conclusion 45 ab §2. π1 46 §3. Some words about open curves. 47 Bibliography 49 Contents CHAPTER 1 Introduction The topological fundamental group can be studied using the theory of covering spaces, since a fundamental group coincides with the group of deck transformations of the asso- ciated universal covering space.
    [Show full text]
  • Abstract Quotients of Profinite Groups, After Nikolov and Segal
    ABSTRACT QUOTIENTS OF PROFINITE GROUPS, AFTER NIKOLOV AND SEGAL BENJAMIN KLOPSCH Abstract. In this expanded account of a talk given at the Oberwolfach Ar- beitsgemeinschaft “Totally Disconnected Groups”, October 2014, we discuss results of Nikolay Nikolov and Dan Segal on abstract quotients of compact Hausdorff topological groups, paying special attention to the class of finitely generated profinite groups. Our primary source is [17]. Sidestepping all difficult and technical proofs, we present a selection of accessible arguments to illuminate key ideas in the subject. 1. Introduction §1.1. Many concepts and techniques in the theory of finite groups depend in- trinsically on the assumption that the groups considered are a priori finite. The theoretical framework based on such methods has led to marvellous achievements, including – as a particular highlight – the classification of all finite simple groups. Notwithstanding, the same methods are only of limited use in the study of infinite groups: it remains mysterious how one could possibly pin down the structure of a general infinite group in a systematic way. Significantly more can be said if such a group comes equipped with additional information, such as a structure-preserving action on a notable geometric object. A coherent approach to studying restricted classes of infinite groups is found by imposing suitable ‘finiteness conditions’, i.e., conditions that generalise the notion of being finite but are significantly more flexible, such as the group being finitely generated or compact with respect to a natural topology. One rather fruitful theme, fusing methods from finite and infinite group theory, consists in studying the interplay between an infinite group Γ and the collection of all its finite quotients.
    [Show full text]
  • 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.
    [Show full text]
  • 24 the Idele Group, Profinite Groups, Infinite Galois Theory
    18.785 Number theory I Fall 2016 Lecture #24 12/06/2015 24 The idele group, profinite groups, infinite Galois theory 24.1 The idele group Let K be a global field. Having introduced the ring of adeles AK in the previous lecture, it is natural to ask about its unit group. As a group we have × × × AK = f(av) 2 AK : av 2 Kv for all v 2 MK ; and av 2 Ov for almost all v 2 MK g; however, as a subspace of AK , this is not a topological group because the inversion map a 7! a−1 need not be continuous. Example 24.1. Consider K = Q and for each prime p let xp = (1; 1;:::; 1; 1; p; 1; 1;:::) 2 AQ be the adele with a p in its pth component and 1's elsewhere. Every basic open set U about 1 in A has the form Q Y Y U = Uv × Ov; v2S V 62S with S ⊆ MQ finite and 1v 2 Uv, and it is clear that U contains xp for all sufficiently large p. It follows that the sequence x2; x3; x5;::: converges to 1 in the topology of AQ. But notice −1 −1 −1 −1 that U does not contain xp for all sufficiently large p, so the sequence x2 ; x3 ; x5 ;::: −1 −1 cannot possibly converge to 1 = 1 in AQ. Thus the function x ! x is not continuous × in the subspace topology for AK . This problem is not specific to rings of adeles. In general, given a topological ring R, there is no reason to expect its unit group R× ⊆ R to be a topological group in the subspace topology unless R happens to be a subring of a topological field (the definition of which × requires inversion to be continuous), as is the case for the unit group OK ; this explains why we have not encountered this problem up to now.
    [Show full text]
  • Subgroups of Finite Index in Profinite Groups
    Subgroups of Finite Index in Profinite Groups Sara Jensen May 14, 2013 1 Introduction In addition to having a group structure, profinite groups have a nontrivial topological structure. Many results pertaining to profinite groups exploit both structures, and there- fore both structures are important in understanding profinite groups. An amazing result due to Nikolov and Segal is the following theorem. Theorem 1.1. Suppose that G is a topologically finitely generated profinite group. Then every subgroup of G of finite index is open. One way to view Theorem 1.1 is as a statement that the algebraic structure of a finitely generated profinite group somehow also encodes the topological structure. That is, if one wishes to know the open subgroups of a profinite group G, a topological property, one must only consider the subgroups of G of finite index, an algebraic property. As profinite groups are compact topological spaces, an open subgroup of G necessarily has finite index. Thus it is also possible to begin with a subgroup of G having a particular topological property (the subgroup is open) and deduce that this subgroup must also have a particular algebraic property (the subgroup has finite index). The proof of Theorem 1.1 is quite extensive, and requires the classification of finite simple groups. However, if one restricts attention to a smaller class of groups, the result can be done in a fairly straightforward manner. Suppose that G is a finite group having a normal series 1 = Gl ⊆ Gl−1 ⊆ ::: ⊆ G1 ⊆ G0 = G such that Gi=Gi+1 is nilpotent for all 0 ≤ i < l.
    [Show full text]
  • Historical Notes on Loop Theory
    Comment.Math.Univ.Carolin. 41,2 (2000)359–370 359 Historical notes on loop theory Hala Orlik Pflugfelder Abstract. This paper deals with the origins and early history of loop theory, summarizing the period from the 1920s through the 1960s. Keywords: quasigroup theory, loop theory, history Classification: Primary 01A60; Secondary 20N05 This paper is an attempt to map, to fit together not only in a geographical and a chronological sense but also conceptually, the various areas where loop theory originated and through which it moved during the early part of its 70 years of history. 70 years is not very much compared to, say, over 300 years of differential calculus. But it is precisely because loop theory is a relatively young subject that it is often misinterpreted. Therefore, it is extremely important for us to acknowledge its distinctive origins. To give an example, when somebody asks, “What is a loop?”, the simplest way to explain is to say, “it is a group without associativity”. This is true, but it is not the whole truth. It is essential to emphasize that loop theory is not just a generalization of group theory but a discipline of its own, originating from and still moving within four basic research areas — algebra, geometry, topology, and combinatorics. Looking back on the first 50 years of loop history, one can see that every decade initiated a new and important phase in its development. These distinct periods can be summarized as follows: I. 1920s the first glimmerings of non-associativity II. 1930s the defining period (Germany) III. 1940s-60s building the basic algebraic frame and new approaches to projective geometry (United States) IV.
    [Show full text]
  • 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 .
    [Show full text]
  • The Equivalence Between Rational G-Sheaves and Rational G-Mackey
    THE EQUIVALENCE BETWEEN RATIONAL G-SHEAVES AND RATIONAL G-MACKEY FUNCTORS FOR PROFINITE G DAVID BARNES AND DANNY SUGRUE Abstract. For G a profinite group, we construct an equivalence between rational G-Mackey functors and a certain full subcategory of G-sheaves over the space of closed subgroups of G called Weyl-G-sheaves. This subcategory consists of those sheaves whose stalk over a subgroup K is K-fixed. This extends the classification of rational G-Mackey functors for finite G of Thévenaz and Webb, and Greenlees and May to a new class of examples. Moreover, this equivalence is instru- mental in the classification of rational G-spectra for profinite G, as given in the second author’s thesis. Contents 1. Introduction 2 1.1. Organisation 3 2. Mackey functors and sheaves 3 2.1. Basic facts on profinite groups 3 2.2. Mackey functors for profinite groups 4 2.3. Burnside ring idempotents 6 2.4. Equivariant sheaves 10 3. The functors 11 3.1. Weyl-G-sheaves determine Mackey Functors 11 3.2. Mackey Functors determine Weyl-G-sheaves 13 4. The equivalence 16 4.1. The equivalence on sheaves 17 4.2. The equivalence on Mackey functors 19 5. Consequences 20 5.1. Examples 20 arXiv:2002.11745v1 [math.AT] 26 Feb 2020 5.2. Weyl Sheaves from equivariant sheaves 21 References 22 1 WEYL SHEAVES AND MACKEY FUNCTORS 2 1. Introduction The classification of rational Mackey functors for finite groups is well-known and highly useful. An algebraic version of this result is given by Thévenaz and Webb [TW95].
    [Show full text]