A geometry of 7-point planes

Save this PDF as:

Size: px
Start display at page:

Transcription

1 A geometry of 7-point planes (Material Based on [Tay09, Chapter 6, pp 43-45]) (Wednesday 16th March 2014) Abstract The classification of finite simple groups gives four families of groups whose union is precisely the set of all finite simple groups. However, when are two such groups isomorphic to one another? In this talk we will show that the alternating group of degree 8 is isomorphic to the matrix group PSL(4,2) by considering group actions on a geometry of 7-point planes. No prior knowledge of geometries is required. 1 Motivation We begin with everyone s favourite mathematical theorem: Theorem 1 (The Classification Theorem of Finite Simple Groups). Every finite simple group is isomorphic to one of the following: (i) A cyclic group of prime order; (ii) A n for some n 5; (iii) A simple group of lie type - these include lots of matrix groups such as projective special linear, unitary, symplectic, or orthogonal groups over a finite field; or (iv) one of 26 sporadic groups. The classification theorem has had a profound effect on finite group theory. Indeed, many important theorems have now been proved using it. However, there is repetition within the theorem. Indeed we have the isomorphisms (see [Wil09] for more details). P SL 2 (4) = P SL2 (5) = Alt(5) P SL 2 (7) = P SL3 (2) P SL 2 (9) = Alt(6) P SL 4 (2) = Alt(8) P SU 4 (2) = P Sp4 (3) In this talk we will prove the penaultimate isomorphism by considering finite geometries. 1

2 2 Finite Geometries We begin with the definition of a geometry. Definition 2. A geometry over a set is a triple (Γ,, τ), where; Γ is a set (the elements of Γ); is a symmetric relation on Γ (the incident relation of Γ); τ : Γ is a surjective map (the type function of Γ). such that if x, y Γ and x y, then τ(x) τ(y). We often talk about the geometry Γ, rather than (Γ,, τ). Geometries arise in many different areas of mathematics. Example 3. Consider a cube, having vertex set V, edge set E and face set F. Defining Γ = V E F, by a b a b or b a and τ : Γ := v, e, f by v if x V ; τx := e if x E; and f if x F, then Γ = (Γ,, τ) is a geometry over. Example 4. Let G be a finite group, p a prime with p G. Let Γ be the set of all chains of p-subgroups of G, and suppose the maximal length of such a chain is m (so this is just the order complex of the Brown complex). Define := {1,..., m} and define the natural map τ : Γ by τ(x) = length(x). By adding the relation defined to be inclusion, we have that Γ = (Γ,, τ) is a geometry over. Example 4 is a specific case of a simplicial complex, and clearly any simplicial complex of dimension d can be considered as a geometry over {1,..., d + 1} in the natural way. We now consider a specific geometry. 2

3 3 7-point planes Let Ω be a set of seven elements, and denote by L the set of 3-element subsets of Ω. Our aim is to identify the elements of L with lines of a projective geometry. Hence we call elements of L Lines (with a capital l!) Definition 5. A 7-point plane is a set B of seven Lines such that any two Lines of B have exactly one element of Ω in common. Definition 6. A fan with vertex α Ω is a set of three Lines such that any pair of Lines have only α in common. We may think of fans diagrammatically as For each α Ω, there are ( 6 2, 2, 2 )/6 = 15 fans on α, and hence there are 105 fans. Moreover, each fan can be extended to a seven point plane in exactly two ways; either Every 7-point plane contains a fan with vertex α, for each α Ω (as there are ( 7 2 ) = 21 common elements of the seven lines in the plane, so 3 pairs of distinct lines each have common element α for a given α Ω). It follows that the set A of 7-point planes on Ω has cardinality 30. Clearly Sym(7) acts on Ω, and hence permutes the 7-point planes. Since the subgroup of Sym(7) which fixes a 7-point plane is P SL 3 (2) (we just assume this) and [Sym(7) : P SL 3 (2)] = 30, we see 1 that the action of Sym(7) on A is transitive. Similarly as P SL 3 (2) Alt(7) and 1 using the Orbit-Stabilizer Theorem, as [Sym(7) : Stab(x)] = Orbit(x) for all x Ω. 3

4 [Alt(7) : P SL 3 (2)] = 15, we see that Alt(7) has two orbits on A, each of length 15. Let P and H denote the two orbits of Alt(7) on A. we call the elements of P and H, Points and Planes respectively. we wish to define a geometry from P, L and H and then prove that these sets are the points, lines and planes of a projective geometry of a vector space of dimension 4 over GF (2). Set := {P, L, H}, Γ = P L H and τ : Γ in the obvious way: P if σ P τ(σ) := L if σ L H if σ H. We then define an incidence relation as follows; a Line λ L is incident to a Point or Plane B P H whenever λ B; the Point B 1 is incident with the Plane B 2 whenever B 1 B 2 is a fan. Thus (Γ,, τ) is a geometry over. To identify the geometry Γ, we proceed to investigate the incident relation in some detail. First we note that Alt(7) acts transitively on each of the sets P, L and H, and preserves (since it merely relabels the elements of Ω). Moreover, the elements of Sym(7) \ Alt(7) interchange P and H. We are now in a position to prove our first result: Lemma 7. (i) Each Line is incident to 3 Points and 3 Planes. (ii) A Point B 1 is incident to a Plane B 2 if and only if B 1 and B 2 are incident to a common line. Proof. (i) Let λ be a Line, and let B A contain λ. Each α λ belongs to two Lines of B other than λ and these Lines create a partition of Ω \ λ into two sets of size 2. Hence we have a bijection between the 3 elements of λ and the 3 partitions of Ω \ λ into two sets of size 2. Conversely, every such bijection arises from a unique 7-point plane which contains λ (to see this just try constructing such a plane). Since there are 6 such bijections (since a bijection is from an set of cardinality 3 to another set of cardinality 3, so is an element of Sym(3)), we deduce that λ is contained in six 7-point planes. Let L denote the stabilizer of λ in Sym(7). Then each element of L permutes the 3 elements of λ, and also permutes the 4 elements of Ω\λ. Thus L = R T, where R = Sym(3) and T = Sym(4). Since L fixes λ, it acts on the six 7-point planes containing λ. Moreover, the transpositions of L interchange the 7-point planes of P with those of H. Thus λ is incident to 3 Points and 3 Planes. (ii) Clearly the given condition is necessary for B 1 and B 2 to be incident. To see that it is also sufficient, assume B 1 and B 2 are incident to a common line λ. By (i), there are three Points and three Planes incident to λ. Moreover, every transposition of L fixes a fan containing λ (possibly 4

5 illustrate this on the board). Each of the 3 transpositions of R maps B 1 to a Plane containing λ, and there are three such Planes, one of which is B 2. Hence B 1 is incident to B 2. We now relate this geometry to a certain projective geometry. Definition 8. If V is a finite-dimensional vector space, then the set of all subspaces of V, partially ordered by inclusion, is called the projective geometry of V, denoted P(V ). Theorem 9. The Points P, Lines L and Planes H can be identified with the points (1-dimensional subspace), lines (2-dimensional subspaces) and planes (3-dimensional subspaces) of the projective geometry of a vector space of dimension 4 over GF (2), so that the incidence relation becomes the usual containment of subspaces. Proof. If B is a Point, there are 7 Lines incident to B. Each of these Lines will be incident to a further two Points. Moreover, any two Points can be incident to at most one common Line (since any two lines of a Point by definition have a common element α, so we can extend these two Lines to a flag at α, and any flag can be extended to a unique Point). Thus we have accounted for all 15 Points. Furthermore, each pair of distinct Points is incident to a unique Line. An analogous argument shows that every pair of Planes are incident to a unique Line. Also, since there are 7 flags in B and each flag is incident to a unique Plane, there are 7 Planes incident to B. Let V = P {0} and let B, B 1, B 2 P (B 1 B 2 ). Define B + B = 0 and 0 + B = B + 0 = B, and define B 1 + B 2 to be the third Point incident to the unique Line determined by B 1 and B 2. Any two Lines with a common Point determine a fan at the common element, and hence are incident to a common Plane, H. This in turn implies associativity of addition, and hence V is a GF (2)-vector space. To see this, we may note that H is incident to 7 Lines and 7 Points, such that each Line is incident to 3 Points and vice versa. Thus representing Points as dots in the following diagram of the Plane H, we see that It is then clear that associativity holds. By construction, the sets P, L and H of all Points, Lines and Planes correspond to the 1-,2- and 3-dimensional subspaces of V, namely the points, lines and planes of the projective geometry of V. 5

6 It is sometimes useful to represent a 7-point plane, B, diagrammatically as: By labeling the elements of Ω as 1,..., 7, we can see that the stabiliser of an incident Point- Line (or Plane-Line) pair has order 24 (since you may permute the elements of the line in any way, and then the Point is uniquely determined by the positioning of one further element of Ω). Since Alt(7) /24 = 105 and there are 15 7 = 105 incident Point-Line (Plane-Line) pairs, it follows that Alt(7) acts transitively on such pairs. Theorem 10. We have Alt(7) P GL 4 (2) and Alt(7) acts 2-transitively on both the points and planes of the projective geometry of dimension 4 ( the book says dimension 3) over GF (2). Proof. We know that Alt(7) acts transitively on P, and two distinct elements of Alt(7) have distinct actions on P. Since the set P of Points correspond to the points of the projective geometry of dimension 4 over GF (2), it follows that Alt(7) P GL 4 (2). Let B i (for i = 1, 2, 3, 4) be Points, and let λ and λ be the lines determined by the pairs (B 1, B 2 ) and (B 3, B 4 ). Since Alt(7) acts transitively on Point-Line pairs, there exists σ Alt(7) such that σ maps (B 1, λ) to (B 3, λ ). Moreover, σ maps B 2 to some Point B 4 incident to λ. The stabilizer in Alt(7) of λ acts as Sym(3) on the Line λ. Hence as there are three Points incident to λ, there exists γ Alt(7) that maps (B 3, B 4 ) to (B 3, B 4 ). It follows that Alt(7) acts 2-transitively on P and hence on the points of the projective geometry. By symmetry, Alt(7) is 2-transitive on H as well. Corollary 11. GL 4 (2) = SL 4 (2) = P SL 4 (2) = P GL 4 (2) = Alt(8). Proof. We know that [P SL 4 (2) : Alt(7)] = 8 ( P SL 4 (2) = 20160, Alt(7) = 2520 and Sym(8) = 40320), and P SL 4 (2) acts on the 8 cosets of Alt(7) in P SL 4 (2). It follows that we may define a homomorphism f : P SL 4 (2) Sym(8). As P SL 4 (2) is simple, ker f is trivial, so we may consider P SL 4 (2) Sym(8). As [P SL 4 (2) : Sym(8)] = 2, and Alt(8) is the only subgroup of Sym(8) of index 2, we must have SL 4 (2) = Alt(8). 6

7 References [Tay09] Donald E. Taylor. The Geometry of the Classical Groups, volume 9 of Sigma Series in Pure Mathematics. Heldermann Verlag, Lemgo, Germany, [Wil09] Robert A. Wilson. The Finite Simple Groups, volume 251 of Graduate Texts in Mathematics. Springer-Verlag, London, United Kingdom,

Group Theory. Contents

Group Theory Contents Chapter 1: Review... 2 Chapter 2: Permutation Groups and Group Actions... 3 Orbits and Transitivity... 6 Specific Actions The Right regular and coset actions... 8 The Conjugation

ON GALOIS REALIZATIONS OF THE 2-COVERABLE SYMMETRIC AND ALTERNATING GROUPS

ON GALOIS REALIZATIONS OF THE 2-COVERABLE SYMMETRIC AND ALTERNATING GROUPS DANIEL RABAYEV AND JACK SONN Abstract. Let f(x) be a monic polynomial in Z[x] with no rational roots but with roots in Q p for

GROUPS ACTING ON A SET

GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

G = G 0 > G 1 > > G k = {e}

Proposition 49. 1. A group G is nilpotent if and only if G appears as an element of its upper central series. 2. If G is nilpotent, then the upper central series and the lower central series have the same

Chapter 7. Permutation Groups

Chapter 7 Permutation Groups () We started the study of groups by considering planar isometries In the previous chapter, we learnt that finite groups of planar isometries can only be cyclic or dihedral

The cover SU(2) SO(3) and related topics

The cover SU(2) SO(3) and related topics Iordan Ganev December 2011 Abstract The subgroup U of unit quaternions is isomorphic to SU(2) and is a double cover of SO(3). This allows a simple computation of

Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson

Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement

Classification of Cartan matrices

Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms

Assignment 8: Selected Solutions

Section 4.1 Assignment 8: Selected Solutions 1. and 2. Express each permutation as a product of disjoint cycles, and identify their parity. (1) (1,9,2,3)(1,9,6,5)(1,4,8,7)=(1,4,8,7,2,3)(5,9,6), odd; (2)

ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP. A. K. Das and R. K. Nath

International Electronic Journal of Algebra Volume 7 (2010) 140-151 ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP A. K. Das and R. K. Nath Received: 12 October 2009; Revised: 15 December

Tiers, Preference Similarity, and the Limits on Stable Partners

Tiers, Preference Similarity, and the Limits on Stable Partners KANDORI, Michihiro, KOJIMA, Fuhito, and YASUDA, Yosuke February 7, 2010 Preliminary and incomplete. Do not circulate. Abstract We consider

ISOMETRIES OF R n KEITH CONRAD

ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible

INCIDENCE-BETWEENNESS GEOMETRY

INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full

Mathematical Induction

Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,

GENERATING SETS KEITH CONRAD 1 Introduction In R n, every vector can be written as a unique linear combination of the standard basis e 1,, e n A notion weaker than a basis is a spanning set: a set of vectors

SOLUTIONS TO ASSIGNMENT 1 MATH 576

SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts

Ideal Class Group and Units

Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals

Chapter 20. Vector Spaces and Bases

Chapter 20. Vector Spaces and Bases In this course, we have proceeded step-by-step through low-dimensional Linear Algebra. We have looked at lines, planes, hyperplanes, and have seen that there is no limit

SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties

SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces

The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

Chapter 7: Products and quotients

Chapter 7: Products and quotients Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 7: Products

Sets of Fibre Homotopy Classes and Twisted Order Parameter Spaces

Communications in Mathematical Physics Manuscript-Nr. (will be inserted by hand later) Sets of Fibre Homotopy Classes and Twisted Order Parameter Spaces Stefan Bechtluft-Sachs, Marco Hien Naturwissenschaftliche

GROUP ACTIONS KEITH CONRAD 1. Introduction The symmetric groups S n, alternating groups A n, and (for n 3) dihedral groups D n behave, by their very definition, as permutations on certain sets. The groups

arxiv:1203.1525v1 [math.co] 7 Mar 2012

Constructing subset partition graphs with strong adjacency and end-point count properties Nicolai Hähnle haehnle@math.tu-berlin.de arxiv:1203.1525v1 [math.co] 7 Mar 2012 March 8, 2012 Abstract Kim defined

On an algorithm for classification of binary self-dual codes with minimum distance four

Thirteenth International Workshop on Algebraic and Combinatorial Coding Theory June 15-21, 2012, Pomorie, Bulgaria pp. 105 110 On an algorithm for classification of binary self-dual codes with minimum

Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs

CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like

Group Fundamentals. Chapter 1. 1.1 Groups and Subgroups. 1.1.1 Definition

Chapter 1 Group Fundamentals 1.1 Groups and Subgroups 1.1.1 Definition A group is a nonempty set G on which there is defined a binary operation (a, b) ab satisfying the following properties. Closure: If

Codes for Network Switches

Codes for Network Switches Zhiying Wang, Omer Shaked, Yuval Cassuto, and Jehoshua Bruck Electrical Engineering Department, California Institute of Technology, Pasadena, CA 91125, USA Electrical Engineering

= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that

Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without

I = 0 1. 1 ad bc. be the set of A in GL(2, C) with real entries and with determinant equal to 1, 1, respectively. Note that A = T A : S S

Fractional linear transformations. Definition. GL(, C) be the set of invertible matrices [ ] a b c d with complex entries. Note that (i) The identity matrix is in GL(, C). [ ] 1 0 I 0 1 (ii) If A and B

PUZZLES WITH POLYHEDRA AND PERMUTATION GROUPS

PUZZLES WITH POLYHEDRA AND PERMUTATION GROUPS JORGE REZENDE. Introduction Consider a polyhedron. For example, a platonic, an arquemidean, or a dual of an arquemidean polyhedron. Construct flat polygonal

Introduction to Finite Fields (cont.)

Chapter 6 Introduction to Finite Fields (cont.) 6.1 Recall Theorem. Z m is a field m is a prime number. Theorem (Subfield Isomorphic to Z p ). Every finite field has the order of a power of a prime number

4. CLASSES OF RINGS 4.1. Classes of Rings class operator A-closed Example 1: product Example 2:

4. CLASSES OF RINGS 4.1. Classes of Rings Normally we associate, with any property, a set of objects that satisfy that property. But problems can arise when we allow sets to be elements of larger sets

EXERCISES FOR THE COURSE MATH 570, FALL 2010

EXERCISES FOR THE COURSE MATH 570, FALL 2010 EYAL Z. GOREN (1) Let G be a group and H Z(G) a subgroup such that G/H is cyclic. Prove that G is abelian. Conclude that every group of order p 2 (p a prime

α = u v. In other words, Orthogonal Projection

Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v

Math 312 Homework 1 Solutions

Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please

4.1 Modules, Homomorphisms, and Exact Sequences

Chapter 4 Modules We always assume that R is a ring with unity 1 R. 4.1 Modules, Homomorphisms, and Exact Sequences A fundamental example of groups is the symmetric group S Ω on a set Ω. By Cayley s Theorem,

POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS

POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS RUSS WOODROOFE 1. Unique Factorization Domains Throughout the following, we think of R as sitting inside R[x] as the constant polynomials (of degree 0).

On the representability of the bi-uniform matroid

On the representability of the bi-uniform matroid Simeon Ball, Carles Padró, Zsuzsa Weiner and Chaoping Xing August 3, 2012 Abstract Every bi-uniform matroid is representable over all sufficiently large

Tree-representation of set families and applications to combinatorial decompositions

Tree-representation of set families and applications to combinatorial decompositions Binh-Minh Bui-Xuan a, Michel Habib b Michaël Rao c a Department of Informatics, University of Bergen, Norway. buixuan@ii.uib.no

COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS

Bull Austral Math Soc 77 (2008), 31 36 doi: 101017/S0004972708000038 COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS IGOR V EROVENKO and B SURY (Received 12 April 2007) Abstract We compute

1 Local Brouwer degree

1 Local Brouwer degree Let D R n be an open set and f : S R n be continuous, D S and c R n. Suppose that the set f 1 (c) D is compact. (1) Then the local Brouwer degree of f at c in the set D is defined.

MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

Introduction to Modern Algebra

Introduction to Modern Algebra David Joyce Clark University Version 0.0.6, 3 Oct 2008 1 1 Copyright (C) 2008. ii I dedicate this book to my friend and colleague Arthur Chou. Arthur encouraged me to write

University of Oslo MAT2 Project The Banach-Tarski Paradox Author: Fredrik Meyer Supervisor: Nadia S. Larsen Abstract In its weak form, the Banach-Tarski paradox states that for any ball in R, it is possible

A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries

Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do

Chapter 13: Basic ring theory

Chapter 3: Basic ring theory Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 3: Basic ring

Recall that two vectors in are perpendicular or orthogonal provided that their dot

Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal

( ) which must be a vector

MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are

GROUP ALGEBRAS. ANDREI YAFAEV

GROUP ALGEBRAS. ANDREI YAFAEV We will associate a certain algebra to a finite group and prove that it is semisimple. Then we will apply Wedderburn s theory to its study. Definition 0.1. Let G be a finite

COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS

COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS IGOR V. EROVENKO AND B. SURY ABSTRACT. We compute commutativity degrees of wreath products A B of finite abelian groups A and B. When B

A NOTE ON TRIVIAL FIBRATIONS

A NOTE ON TRIVIAL FIBRATIONS Petar Pavešić Fakulteta za Matematiko in Fiziko, Univerza v Ljubljani, Jadranska 19, 1111 Ljubljana, Slovenija petar.pavesic@fmf.uni-lj.si Abstract We study the conditions

ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS

ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS John A. Beachy Northern Illinois University 2014 ii J.A.Beachy This is a supplement to Abstract Algebra, Third Edition by John A. Beachy and William D. Blair

Department of Pure Mathematics and Mathematical Statistics University of Cambridge GEOMETRY AND GROUPS

Department of Pure Mathematics and Mathematical Statistics University of Cambridge GEOMETRY AND GROUPS Albrecht Dürer s engraving Melencolia I (1514) Notes Michaelmas 2012 T. K. Carne. t.k.carne@dpmms.cam.ac.uk

SOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS

SOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS Ivan Kolář Abstract. Let F be a fiber product preserving bundle functor on the category FM m of the proper base order r. We deduce that the r-th

FIBER PRODUCTS AND ZARISKI SHEAVES

FIBER PRODUCTS AND ZARISKI SHEAVES BRIAN OSSERMAN 1. Fiber products and Zariski sheaves We recall the definition of a fiber product: Definition 1.1. Let C be a category, and X, Y, Z objects of C. Fix also

Linear Algebra I. Ronald van Luijk, 2012

Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.

THE AVERAGE DEGREE OF AN IRREDUCIBLE CHARACTER OF A FINITE GROUP

THE AVERAGE DEGREE OF AN IRREDUCIBLE CHARACTER OF A FINITE GROUP by I. M. Isaacs Mathematics Department University of Wisconsin 480 Lincoln Dr. Madison, WI 53706 USA E-Mail: isaacs@math.wisc.edu Maria

SOLUTIONS TO PROBLEM SET 3

SOLUTIONS TO PROBLEM SET 3 MATTI ÅSTRAND The General Cubic Extension Denote L = k(α 1, α 2, α 3 ), F = k(a 1, a 2, a 3 ) and K = F (α 1 ). The polynomial f(x) = x 3 a 1 x 2 + a 2 x a 3 = (x α 1 )(x α 2

Fundamentals of Media Theory

Fundamentals of Media Theory ergei Ovchinnikov Mathematics Department an Francisco tate University an Francisco, CA 94132 sergei@sfsu.edu Abstract Media theory is a new branch of discrete applied mathematics

ON FIBER DIAMETERS OF CONTINUOUS MAPS

ON FIBER DIAMETERS OF CONTINUOUS MAPS PETER S. LANDWEBER, EMANUEL A. LAZAR, AND NEEL PATEL Abstract. We present a surprisingly short proof that for any continuous map f : R n R m, if n > m, then there

SMALL SKEW FIELDS CÉDRIC MILLIET

SMALL SKEW FIELDS CÉDRIC MILLIET Abstract A division ring of positive characteristic with countably many pure types is a field Wedderburn showed in 1905 that finite fields are commutative As for infinite

ON THE DEGREE OF MAXIMALITY OF DEFINITIONALLY COMPLETE LOGICS

Bulletin of the Section of Logic Volume 15/2 (1986), pp. 72 79 reedition 2005 [original edition, pp. 72 84] Ryszard Ladniak ON THE DEGREE OF MAXIMALITY OF DEFINITIONALLY COMPLETE LOGICS The following definition

Fiber Polytopes for the Projections between Cyclic Polytopes

Europ. J. Combinatorics (2000) 21, 19 47 Article No. eujc.1999.0319 Available online at http://www.idealibrary.com on Fiber Polytopes for the Projections between Cyclic Polytopes CHRISTOS A. ATHANASIADIS,

Math 231b Lecture 35. G. Quick

Math 231b Lecture 35 G. Quick 35. Lecture 35: Sphere bundles and the Adams conjecture 35.1. Sphere bundles. Let X be a connected finite cell complex. We saw that the J-homomorphism could be defined by

ORIENTATIONS. Contents

ORIENTATIONS Contents 1. Generators for H n R n, R n p 1 1. Generators for H n R n, R n p We ended last time by constructing explicit generators for H n D n, S n 1 by using an explicit n-simplex which

Linear Authentication Codes: Bounds and Constructions

866 IEEE TRANSACTIONS ON INFORMATION TNEORY, VOL 49, NO 4, APRIL 2003 Linear Authentication Codes: Bounds and Constructions Huaxiong Wang, Chaoping Xing, and Rei Safavi-Naini Abstract In this paper, we

Chapter 9 Unitary Groups and SU(N)

Chapter 9 Unitary Groups and SU(N) The irreducible representations of SO(3) are appropriate for describing the degeneracies of states of quantum mechanical systems which have rotational symmetry in three

Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

Singular fibers of stable maps and signatures of 4 manifolds

359 399 359 arxiv version: fonts, pagination and layout may vary from GT published version Singular fibers of stable maps and signatures of 4 manifolds OSAMU SAEKI TAKAHIRO YAMAMOTO We show that for a

1 The Line vs Point Test

6.875 PCP and Hardness of Approximation MIT, Fall 2010 Lecture 5: Low Degree Testing Lecturer: Dana Moshkovitz Scribe: Gregory Minton and Dana Moshkovitz Having seen a probabilistic verifier for linearity

Orthogonal Projections

Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors

CONTRIBUTIONS TO ZERO SUM PROBLEMS

CONTRIBUTIONS TO ZERO SUM PROBLEMS S. D. ADHIKARI, Y. G. CHEN, J. B. FRIEDLANDER, S. V. KONYAGIN AND F. PAPPALARDI Abstract. A prototype of zero sum theorems, the well known theorem of Erdős, Ginzburg

THREE DIMENSIONAL GEOMETRY

Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,

E3: PROBABILITY AND STATISTICS lecture notes

E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................

Group Theory. Chapter 1

Chapter 1 Group Theory Most lectures on group theory actually start with the definition of what is a group. It may be worth though spending a few lines to mention how mathematicians came up with such a

Math/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability

Math/Stats 425 Introduction to Probability 1. Uncertainty and the axioms of probability Processes in the real world are random if outcomes cannot be predicted with certainty. Example: coin tossing, stock

Row Ideals and Fibers of Morphisms

Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion

1 VECTOR SPACES AND SUBSPACES

1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such

1 = (a 0 + b 0 α) 2 + + (a m 1 + b m 1 α) 2. for certain elements a 0,..., a m 1, b 0,..., b m 1 of F. Multiplying out, we obtain

Notes on real-closed fields These notes develop the algebraic background needed to understand the model theory of real-closed fields. To understand these notes, a standard graduate course in algebra is

3. Linear Programming and Polyhedral Combinatorics

Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the

Lecture 4 DISCRETE SUBGROUPS OF THE ISOMETRY GROUP OF THE PLANE AND TILINGS

1 Lecture 4 DISCRETE SUBGROUPS OF THE ISOMETRY GROUP OF THE PLANE AND TILINGS This lecture, just as the previous one, deals with a classification of objects, the original interest in which was perhaps

TOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS

TOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS VIPUL NAIK Abstract. In this journey, we are going to explore the so called separation axioms in greater detail. We shall try to understand how these axioms

Cacti with minimum, second-minimum, and third-minimum Kirchhoff indices

MATHEMATICAL COMMUNICATIONS 47 Math. Commun., Vol. 15, No. 2, pp. 47-58 (2010) Cacti with minimum, second-minimum, and third-minimum Kirchhoff indices Hongzhuan Wang 1, Hongbo Hua 1, and Dongdong Wang

Cameron Liebler line classes

Cameron Liebler line classes Alexander Gavrilyuk Krasovsky Institute of Mathematics and Mechanics, (Yekaterinburg, Russia) based on joint work with Ivan Mogilnykh, Institute of Mathematics (Novosibirsk,

Notes on finite group theory. Peter J. Cameron

Notes on finite group theory Peter J. Cameron October 2013 2 Preface Group theory is a central part of modern mathematics. Its origins lie in geometry (where groups describe in a very detailed way the

Linear Codes. Chapter 3. 3.1 Basics

Chapter 3 Linear Codes In order to define codes that we can encode and decode efficiently, we add more structure to the codespace. We shall be mainly interested in linear codes. A linear code of length

MATH1231 Algebra, 2015 Chapter 7: Linear maps

MATH1231 Algebra, 2015 Chapter 7: Linear maps A/Prof. Daniel Chan School of Mathematics and Statistics University of New South Wales danielc@unsw.edu.au Daniel Chan (UNSW) MATH1231 Algebra 1 / 43 Chapter

DEGREES OF ORDERS ON TORSION-FREE ABELIAN GROUPS

DEGREES OF ORDERS ON TORSION-FREE ABELIAN GROUPS ASHER M. KACH, KAREN LANGE, AND REED SOLOMON Abstract. We construct two computable presentations of computable torsion-free abelian groups, one of isomorphism

o-minimality and Uniformity in n 1 Graphs

o-minimality and Uniformity in n 1 Graphs Reid Dale July 10, 2013 Contents 1 Introduction 2 2 Languages and Structures 2 3 Definability and Tame Geometry 4 4 Applications to n 1 Graphs 6 5 Further Directions

Notes on Group Theory

Notes on Group Theory Mark Reeder March 7, 2014 Contents 1 Notation for sets and functions 4 2 Basic group theory 4 2.1 The definition of a group................................. 4 2.2 Group homomorphisms..................................

ISU Department of Mathematics. Graduate Examination Policies and Procedures

ISU Department of Mathematics Graduate Examination Policies and Procedures There are four primary criteria to be used in evaluating competence on written or oral exams. 1. Knowledge Has the student demonstrated

Inner Product Spaces and Orthogonality

Inner Product Spaces and Orthogonality week 3-4 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,

SHORT CYCLE COVERS OF GRAPHS WITH MINIMUM DEGREE THREE

SHOT YLE OVES OF PHS WITH MINIMUM DEEE THEE TOMÁŠ KISE, DNIEL KÁL, END LIDIKÝ, PVEL NEJEDLÝ OET ŠÁML, ND bstract. The Shortest ycle over onjecture of lon and Tarsi asserts that the edges of every bridgeless

FIXED POINT SETS OF FIBER-PRESERVING MAPS

FIXED POINT SETS OF FIBER-PRESERVING MAPS Robert F. Brown Department of Mathematics University of California Los Angeles, CA 90095 e-mail: rfb@math.ucla.edu Christina L. Soderlund Department of Mathematics