DOCSLIB.ORG

The Automorphism Groups on the Complex Plane Aron Persson

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Automorphism Groups on the Complex Plane Aron Persson The Automorphism Groups on the Complex Plane Aron Persson VT 2017 Essay, 15hp Bachelor in Mathematics, 180hp Department of Mathematics and Mathematical Statistics Abstract The automorphism groups in the complex plane are defined, and we prove that they satisfy the group axioms. The automorphism group is derived for some domains. By applying the Riemann mapping theorem, it is proved that every automorphism group on simply connected domains that are proper subsets of the complex plane, is isomorphic to the automorphism group on the unit disc. Sammanfattning Automorfigrupperna i det komplexa talplanet definieras och vi bevisar att de uppfyller gruppaxiomen. Automorfigruppen på några domän härleds. Genom att applicera Riemanns avbildningssats bevisas att varje automorfigrupp på enkelt sammanhängande, öppna och äkta delmängder av det komplexa talplanet är isomorf med automorfigruppen på enhetsdisken. Contents 1. Introduction 1 2. Preliminaries 3 2.1. Group and Set Theory 3 2.2. Topology and Complex Analysis 5 3. The Definition of the Automorphism Group 9 4. Automorphism Groups on Sets in the Complex Plane 17 4.1. The Automorphism Group on the Complex Plane 17 4.2. The Automorphism Group on the Extended Complex Plane 21 4.3. The Automorphism Group on the Unit Disc 23 4.4. Automorphism Groups and Biholomorphic Mappings 25 4.5. Automorphism Groups on Punctured Domains 28 5. Automorphism Groups and the Riemann Mapping Theorem 35 6. Acknowledgements 43 7. References 45 1. Introduction To study the geometry of complex domains one central tool is the automorphism group. Let D ⊆ C be an open and connected set. The automorphism group of D is then defined as the collection of all biholomorphic mappings D → D together with the composition operator. Loosely, automorphic functions are functions that preserve the structure of a object. The young Poincaré was one of the first to do work on automorphic functions, and ultimately it helped clarify a many different areas of mathematics. He introduced a language by combining different areas of mathematics, and his work on automorphic functions have most famously led to the proof of Fermat´s last theorem. [11] It began as the Academy of Sciences of Paris proposed a challenge in 1880 to perfect some of the language in differential equations in one variable. As Poincaré entered his second memoir in the 1880s, he built the first groundwork on automorphic functions [2]. This memoir focused on Fuchsian functions which was inspired from an article by L. Fuchs. It was Klein who coined the modern term automorphic functions in 1890 [11]. Later, from all the work on automorphic functions, Poincaré and Koebe managed to prove the uniformization theorem, it lead further to the proof that the generalization of the uniformization theorem to arbitrary dimensions does not hold [4]. In Section 4, we shall explicitly deduce the following automorphism groups. Later in this introduction, we will see the importance of knowing explicitly what these groups are. • Aut(C) = {az + b : a, b ∈ C, a 6= 0} (Theorem 4.9); az + b • Aut(ˆ) = : a, b, c, d ∈ , and ad − bc 6= 0 , C cz + d C where Cˆ = C ∪ {∞} (Theorem 4.11); a − z • Aut(B (0)) = eiφ : a ∈ B (0), and φ ∈ (0, 2π] , 1 1 − az¯ 1 where B1(0) ⊂ C is the unit disc (Theorem 4.13); •{ f ∈ Aut(D): f(A) = A} and Aut(D\A) are group isomorphic where D is a bounded domain, and A ⊂ D (Theorem 4.19). The main result of this essay is the following theorem. Theorem 5.10. Let D ⊂ C be a simply connected and open set not equal C. Then Aut(D) is isomorphic to Aut(B1(0)) in the sense of groups. Theorem 5.10 is a direct consequence of the celebrated Riemann mapping theorem (The- orem 5.8). This emphasises the importance of the above examples. Consider a simply connected domain D ⊆ C, then from Theorem 5.10, we need only be interested in Aut(C), and Aut(B1(0)). The domain B1(0)\{0} is not simply connected, and Aut(B1(0)\{0}) is not isomorphic to Aut(B1(0)), which shows that Theorem 5.10 does not hold for connected and open sets that are not simply connected. Even so, it is possible to generalise Theorem 5.10 to Riemann surfaces, i.e. 1-dimensional complex manifolds. The uniformization theorem, as it is called, states: A simply connected Riemann surface, S, is biholomorphically equivalent 1 ˆ to either C , C or B1(0) (see e.g. [1]). In the early 1900 Poincaré showed that Theorem 5.10 is not possible to generalise to higher dimension with the following famous example. He showed that the automorphism group of the bi-disc in C2, 2 {(z1, z2) ∈ C : |z1| < 1, |z2| < 1} , and automorphism group of the unit ball in C2, 2 2 2 {(z1, z2) ∈ C : |z1| + |z2| < 1} . is not isomorphic. Even as they both are simply connected [6]. The overview of the essay is as follows. In Section 2, we introduce some background the- ory. The background related to abstract algebra is based on [3], and the background related to complex analysis is mainly inspired by [8, 10]. In the third Section, the automorphism group is defined. Ultimately, by some theorems taken from [8], the group axioms of the automorphism group are proved. In Section four, the automorphism group on the complex plane, extended complex plane and unit disc is derived. Also, biholomorphic equivalence is introduced, and automorphism groups on punctured domains are considered. Mainly, this section is inspired by [4–6,8]. In Section 5, Theorem 5.10 is proved by applying the Riemann mapping theorem, this section is based on [5, 6]. 2 2. Preliminaries In this section, we shall introduce some concepts from group theory and complex analysis that will be needed in this essay. 2.1. Group and Set Theory. For our purposes we will take the notion of a set to be obvious, it is well understood what a collection of something is. Thus we take the definition of a set intuitively. However, the concept of supremum is very useful, but since not all readers may have encountered it before, we shall state it here. Definition 2.1. Let X ⊂ R, then if there exists a a = min{α ∈ R|x ≤ α, for all x ∈ X} then we say that a is the supremum of X. We denote this as sup(X). Notice that this is not the general definition of supremum on ordered sets, a see e.g. Definition 1.8 in [9]. For our purposes it is sufficient to assume that we take the supremum of some subset of R. In many cases the supremum will be analogous to the maximum of a set, and it may help the reader to intuitively read maximum whenever we talk about supremum. Though, remember they are not always the same. Throughout this essay, we will make frequent use of discs centered at a point, c ∈ C, with radius r > 0, and the one point compactification of the complex plane (or extended complex ˆ plane) . We will denote them as Br(c) and C := C ∪ {∞}, respectively. Furthermore, we denote the unit disc as B1(0). Now, we will briefly go through some fundamental properties from group theory. Loosely speaking group theory is about generalising algebra. There is no need to limit oneself to only talk about numbers and addition, etc. What if we were to replace all the integers with different kinds of bananas (now imagine there are countably infinite amount of different bananas), then we could simply talk about putting together two different bananas and you would potentially get a new kind banana. For these bananas we can not apply directly our regular addition operator, rather in generality we talk about binary operators. Our goal is then to apply binary operators to sets, as in our example, the set of different kinds of bananas. Definition 2.2. A binary operator, ∗, on a set A is a mapping from A × A into A. For (a, b) ∈ A × A, we denote ∗((a, b)) as a ∗ b. Both ordinary addition and multiplication are examples of binary operators on Z. In particular, it is important for our purposes to realise that the composition operator, ◦, is a binary operator on the set of all complex valued functions. Definition 2.3. Let B be a subset of A, and ∗ be a binary operator on A. Then B is said to be closed under ∗, if for every a, b ∈ B, a ∗ b ∈ B. It is easy to check that ◦ is closed under the set of all complex valued functions. By restricting the binary operator to three properties, we can define a so called group. Definition 2.4. Let A be a set closed under a binary operator ∗. We call (A, ∗) a group if: G1: the binary operator, ∗, satisfies a ∗ (b ∗ c) = (a ∗ b) ∗ c, 3 for all a, b, c ∈ A, i.e. ∗ is associative in A. G2: there exists an identity element e ∈ A for ∗, such that e ∗ a = a ∗ e = a, for all a ∈ A; G3: for each a ∈ A there exists an inverse element a0 ∈ A such that a ∗ a0 = a0 ∗ a = e. The definition of a group may seem redundant for the reader with little experience with group theory, but it is really useful. One such use is when we can relate two different groups, we want a mapping between groups that preserves some structure. More formally, we call such mappings group homomorphisms, or simply homomorphisms if there is no risk for ambiguity. Definition 2.5. A mapping φ : G → G0 between two groups (G, ∗), (G0, ∗0) is called a homomorphism if 0 φ(g1 ∗ g2) = φ(g1) ∗ φ(g2) for all g1, g2 ∈ G.
Load more

Recommended publications
  • On Automorphisms and Endomorphisms of Projective Varieties
    On automorphisms and endomorphisms of projective varieties Michel Brion Abstract We first show that any connected algebraic group over a perfect field is the neutral component of the automorphism group scheme of some normal pro- jective variety. Then we show that very few connected algebraic semigroups can be realized as endomorphisms of some projective variety X, by describing the structure of all connected subsemigroup schemes of End(X). Key words: automorphism group scheme, endomorphism semigroup scheme MSC classes: 14J50, 14L30, 20M20 1 Introduction and statement of the results By a result of Winkelmann (see [22]), every connected real Lie group G can be realized as the automorphism group of some complex Stein manifold X, which may be chosen complete, and hyperbolic in the sense of Kobayashi. Subsequently, Kan showed in [12] that we may further assume dimC(X) = dimR(G). We shall obtain a somewhat similar result for connected algebraic groups. We first introduce some notation and conventions, and recall general results on automorphism group schemes. Throughout this article, we consider schemes and their morphisms over a fixed field k. Schemes are assumed to be separated; subschemes are locally closed unless mentioned otherwise. By a point of a scheme S, we mean a Michel Brion Institut Fourier, Universit´ede Grenoble B.P. 74, 38402 Saint-Martin d'H`eresCedex, France e-mail: [email protected] 1 2 Michel Brion T -valued point f : T ! S for some scheme T .A variety is a geometrically integral scheme of finite type. We shall use [17] as a general reference for group schemes.
    [Show full text]
  • Orbits of Automorphism Groups of Fields
    ORBITS OF AUTOMORPHISM GROUPS OF FIELDS KIRAN S. KEDLAYA AND BJORN POONEN Abstract. We address several specific aspects of the following general question: can a field K have so many automorphisms that the action of the automorphism group on the elements of K has relatively few orbits? We prove that any field which has only finitely many orbits under its automorphism group is finite. We extend the techniques of that proof to approach a broader conjecture, which asks whether the automorphism group of one field over a subfield can have only finitely many orbits on the complement of the subfield. Finally, we apply similar methods to analyze the field of Mal'cev-Neumann \generalized power series" over a base field; these form near-counterexamples to our conjecture when the base field has characteristic zero, but often fall surprisingly far short in positive characteristic. Can an infinite field K have so many automorphisms that the action of the automorphism group on the elements of K has only finitely many orbits? In Section 1, we prove that the answer is \no" (Theorem 1.1), even though the corresponding answer for division rings is \yes" (see Remark 1.2). Our proof constructs a \trace map" from the given field to a finite field, and exploits the peculiar combination of additive and multiplicative properties of this map. Section 2 attempts to prove a relative version of Theorem 1.1, by considering, for a non- trivial extension of fields k ⊂ K, the action of Aut(K=k) on K. In this situation each element of k forms an orbit, so we study only the orbits of Aut(K=k) on K − k.
    [Show full text]
  • The Pure Symmetric Automorphisms of a Free Group Form a Duality Group
    THE PURE SYMMETRIC AUTOMORPHISMS OF A FREE GROUP FORM A DUALITY GROUP NOEL BRADY, JON MCCAMMOND, JOHN MEIER, AND ANDY MILLER Abstract. The pure symmetric automorphism group of a finitely generated free group consists of those automorphisms which send each standard generator to a conjugate of itself. We prove that these groups are duality groups. 1. Introduction Let Fn be a finite rank free group with fixed free basis X = fx1; : : : ; xng. The symmetric automorphism group of Fn, hereafter denoted Σn, consists of those au- tomorphisms that send each xi 2 X to a conjugate of some xj 2 X. The pure symmetric automorphism group, denoted PΣn, is the index n! subgroup of Σn of symmetric automorphisms that send each xi 2 X to a conjugate of itself. The quotient of PΣn by the inner automorphisms of Fn will be denoted OPΣn. In this note we prove: Theorem 1.1. The group OPΣn is a duality group of dimension n − 2. Corollary 1.2. The group PΣn is a duality group of dimension n − 1, hence Σn is a virtual duality group of dimension n − 1. (In fact we establish slightly more: the dualizing module in both cases is -free.) Corollary 1.2 follows immediately from Theorem 1.1 since Fn is a 1-dimensional duality group, there is a short exact sequence 1 ! Fn ! PΣn ! OPΣn ! 1 and any duality-by-duality group is a duality group whose dimension is the sum of the dimensions of its constituents (see Theorem 9.10 in [2]). That the virtual cohomological dimension of Σn is n − 1 was previously estab- lished by Collins in [9].
    [Show full text]
  • Riemann Surfaces
    RIEMANN SURFACES AARON LANDESMAN CONTENTS 1. Introduction 2 2. Maps of Riemann Surfaces 4 2.1. Defining the maps 4 2.2. The multiplicity of a map 4 2.3. Ramification Loci of maps 6 2.4. Applications 6 3. Properness 9 3.1. Definition of properness 9 3.2. Basic properties of proper morphisms 9 3.3. Constancy of degree of a map 10 4. Examples of Proper Maps of Riemann Surfaces 13 5. Riemann-Hurwitz 15 5.1. Statement of Riemann-Hurwitz 15 5.2. Applications 15 6. Automorphisms of Riemann Surfaces of genus ≥ 2 18 6.1. Statement of the bound 18 6.2. Proving the bound 18 6.3. We rule out g(Y) > 1 20 6.4. We rule out g(Y) = 1 20 6.5. We rule out g(Y) = 0, n ≥ 5 20 6.6. We rule out g(Y) = 0, n = 4 20 6.7. We rule out g(C0) = 0, n = 3 20 6.8. 21 7. Automorphisms in low genus 0 and 1 22 7.1. Genus 0 22 7.2. Genus 1 22 7.3. Example in Genus 3 23 Appendix A. Proof of Riemann Hurwitz 25 Appendix B. Quotients of Riemann surfaces by automorphisms 29 References 31 1 2 AARON LANDESMAN 1. INTRODUCTION In this course, we’ll discuss the theory of Riemann surfaces. Rie- mann surfaces are a beautiful breeding ground for ideas from many areas of math. In this way they connect seemingly disjoint fields, and also allow one to use tools from different areas of math to study them.
    [Show full text]
  • Generalized Quaternions
    GENERALIZED QUATERNIONS KEITH CONRAD 1. introduction The quaternion group Q8 is one of the two non-abelian groups of size 8 (up to isomor- phism). The other one, D4, can be constructed as a semi-direct product: ∼ ∼ × ∼ D4 = Aff(Z=(4)) = Z=(4) o (Z=(4)) = Z=(4) o Z=(2); where the elements of Z=(2) act on Z=(4) as the identity and negation. While Q8 is not a semi-direct product, it can be constructed as the quotient group of a semi-direct product. We will see how this is done in Section2 and then jazz up the construction in Section3 to make an infinite family of similar groups with Q8 as the simplest member. In Section4 we will compare this family with the dihedral groups and see how it fits into a bigger picture. 2. The quaternion group from a semi-direct product The group Q8 is built out of its subgroups hii and hji with the overlapping condition i2 = j2 = −1 and the conjugacy relation jij−1 = −i = i−1. More generally, for odd a we have jaij−a = −i = i−1, while for even a we have jaij−a = i. We can combine these into the single formula a (2.1) jaij−a = i(−1) for all a 2 Z. These relations suggest the following way to construct the group Q8. Theorem 2.1. Let H = Z=(4) o Z=(4), where (a; b)(c; d) = (a + (−1)bc; b + d); ∼ The element (2; 2) in H has order 2, lies in the center, and H=h(2; 2)i = Q8.
    [Show full text]
  • On the Group-Theoretic Properties of the Automorphism Groups of Various Graphs
    ON THE GROUP-THEORETIC PROPERTIES OF THE AUTOMORPHISM GROUPS OF VARIOUS GRAPHS CHARLES HOMANS Abstract. In this paper we provide an introduction to the properties of one important connection between the theories of groups and graphs, that of the group formed by the automorphisms of a given graph. We provide examples of important results in graph theory that can be understood through group theory and vice versa, and conclude with a treatment of Frucht's theorem. Contents 1. Introduction 1 2. Fundamental Definitions, Concepts, and Theorems 2 3. Example 1: The Orbit-Stabilizer Theorem and its Application to Graph Automorphisms 4 4. Example 2: On the Automorphism Groups of the Platonic Solid Skeleton Graphs 4 5. Example 3: A Tight Bound on the Product of the Chromatic Number and Independence Number of Vertex-Transitive Graphs 6 6. Frucht's Theorem 7 7. Acknowledgements 9 8. References 9 1. Introduction Groups and graphs are two highly important kinds of structures studied in math- ematics. Interestingly, the theory of groups and the theory of graphs are deeply connected. In this paper, we examine one particular such connection: that which emerges from the observation that the automorphisms of any given graph form a group under composition. In section 2, we provide a framework for understanding the material discussed in the paper. In sections 3, 4, and 5, we demonstrate how important results in group theory illuminate some properties of automorphism groups, how the geo- metric properties of particular embeddings of graphs can be used to determine the structure of the automorphism groups of all embeddings of those graphs, and how the automorphism group can be used to determine fundamental truths about the structure of the graph.
    [Show full text]
  • The Centralizer of a Group Automorphism
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 54, 27-41 (1978) The Centralizer of a Group Automorphism CARLTON J. MAXSON AND KIRBY C. SWTH Texas A& M University, College Station, Texas 77843 Communicated by A. Fr6hlich Received May 31, 1971 Let G be a finite group. The structure of the near-ring C(A) of identity preserving functionsf : G + G, which commute with a given automorphism A of G, is investigated. The results are then applied to the case in which G is a finite vector space and A is an invertible linear transformation. Let A be a linear transformation on a finite-dimensional vector space V over a field F. The problem of determining the structure of the ring of linear trans- formations on V which commute with A has been studied extensively (e.g., [3, 5, 81). In this paper we consider a nonlinear analogue to this well-known problem which arises naturally in the study of automorphisms of a linear auto- maton [6]. Specifically let G be a finite group, written additively, and let A be an auto- morphism of G. If C(A) = {f: G + G j fA = Af and f (0) = 0 where 0 is the identity of G), then C(A) forms a near ring under the operations of pointwise addition and function composition. That is (C(A), +) is a group, (C(A), .) is a monoid, (f+-g)h=fh+gh for all f, g, hcC(A) and f.O=O.f=O, f~ C(A).
    [Show full text]
  • Homomorphisms and Isomorphisms
    Lecture 4.1: Homomorphisms and isomorphisms Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Modern Algebra M. Macauley (Clemson) Lecture 4.1: Homomorphisms and isomorphisms Math 4120, Modern Algebra 1 / 13 Motivation Throughout the course, we've said things like: \This group has the same structure as that group." \This group is isomorphic to that group." However, we've never really spelled out the details about what this means. We will study a special type of function between groups, called a homomorphism. An isomorphism is a special type of homomorphism. The Greek roots \homo" and \morph" together mean \same shape." There are two situations where homomorphisms arise: when one group is a subgroup of another; when one group is a quotient of another. The corresponding homomorphisms are called embeddings and quotient maps. Also in this chapter, we will completely classify all finite abelian groups, and get a taste of a few more advanced topics, such as the the four \isomorphism theorems," commutators subgroups, and automorphisms. M. Macauley (Clemson) Lecture 4.1: Homomorphisms and isomorphisms Math 4120, Modern Algebra 2 / 13 A motivating example Consider the statement: Z3 < D3. Here is a visual: 0 e 0 7! e f 1 7! r 2 2 1 2 7! r r2f rf r2 r The group D3 contains a size-3 cyclic subgroup hri, which is identical to Z3 in structure only. None of the elements of Z3 (namely 0, 1, 2) are actually in D3. When we say Z3 < D3, we really mean is that the structure of Z3 shows up in D3.
    [Show full text]
  • Uniformization of Riemann Surfaces Revisited
    UNIFORMIZATION OF RIEMANN SURFACES REVISITED CIPRIANA ANGHEL AND RARES¸STAN ABSTRACT. We give an elementary and self-contained proof of the uniformization theorem for non-compact simply-connected Riemann surfaces. 1. INTRODUCTION Paul Koebe and shortly thereafter Henri Poincare´ are credited with having proved in 1907 the famous uniformization theorem for Riemann surfaces, arguably the single most important result in the whole theory of analytic functions of one complex variable. This theorem generated con- nections between different areas and lead to the development of new fields of mathematics. After Koebe, many proofs of the uniformization theorem were proposed, all of them relying on a large body of topological and analytical prerequisites. Modern authors [6], [7] use sheaf cohomol- ogy, the Runge approximation theorem, elliptic regularity for the Laplacian, and rather strong results about the vanishing of the first cohomology group of noncompact surfaces. A more re- cent proof with analytic flavour appears in Donaldson [5], again relying on many strong results, including the Riemann-Roch theorem, the topological classification of compact surfaces, Dol- beault cohomology and the Hodge decomposition. In fact, one can hardly find in the literature a self-contained proof of the uniformization theorem of reasonable length and complexity. Our goal here is to give such a minimalistic proof. Recall that a Riemann surface is a connected complex manifold of dimension 1, i.e., a connected Hausdorff topological space locally homeomorphic to C, endowed with a holomorphic atlas. Uniformization theorem (Koebe [9], Poincare´ [15]). Any simply-connected Riemann surface is biholomorphic to either the complex plane C, the open unit disk D, or the Riemann sphere C^.
    [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]
  • Isomorphisms and Automorphisms
    Isomorphisms and Automorphisms Definition As usual an isomorphism is defined as a map between objects that preserves structure, for general designs this means: Suppose (X,A) and (Y,B) are two designs. They are said to be isomorphic if there exists a bijection α: X → Y such that if we apply α to the elements of any block of A we obtain a block of B and all blocks of B are obtained this way. Symbolically we can write this as: B = [ {α(x) | x ϵ A} : A ϵ A ]. The bijection α is called an isomorphism. Example We've seen two versions of a (7,4,2) symmetric design: k = 4: {3,5,6,7} {4,6,7,1} {5,7,1,2} {6,1,2,3} {7,2,3,4} {1,3,4,5} {2,4,5,6} Blocks: y = [1,2,3,4] {1,2} = [1,2,5,6] {1,3} = [1,3,6,7] {1,4} = [1,4,5,7] {2,3} = [2,3,5,7] {2,4} = [2,4,6,7] {3,4} = [3,4,5,6] α:= (2576) [1,2,3,4] → {1,5,3,4} [1,2,5,6] → {1,5,7,2} [1,3,6,7] → {1,3,2,6} [1,4,5,7] → {1,4,7,2} [2,3,5,7] → {5,3,7,6} [2,4,6,7] → {5,4,2,6} [3,4,5,6] → {3,4,7,2} Incidence Matrices Since the incidence matrix is equivalent to the design there should be a relationship between matrices of isomorphic designs.
    [Show full text]
  • Automorphisms of Group Extensions
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 155, Number t, March 1971 AUTOMORPHISMS OF GROUP EXTENSIONS BY CHARLES WELLS Abstract. If 1 ->■ G -í> E^-Tt -> 1 is a group extension, with i an inclusion, any automorphism <j>of E which takes G onto itself induces automorphisms t on G and a on n. However, for a pair (a, t) of automorphism of n and G, there may not be an automorphism of E inducing the pair. Let à: n —*■Out G be the homomorphism induced by the given extension. A pair (a, t) e Aut n x Aut G is called compatible if a fixes ker á, and the automorphism induced by a on Hü is the same as that induced by the inner automorphism of Out G determined by t. Let C< Aut IT x Aut G be the group of compatible pairs. Let Aut (E; G) denote the group of automorphisms of E fixing G. The main result of this paper is the construction of an exact sequence 1 -» Z&T1,ZG) -* Aut (E; G)-+C^ H*(l~l,ZG). The last map is not surjective in general. It is not even a group homomorphism, but the sequence is nevertheless "exact" at C in the obvious sense. 1. Notation. If G is a group with subgroup H, we write H<G; if H is normal in G, H<¡G. CGH and NGH are the centralizer and normalizer of H in G. Aut G, Inn G, Out G, and ZG are the automorphism group, the inner automorphism group, the outer automorphism group, and the center of G, respectively.
    [Show full text]