COMBINATORIAL GROUP THEORY NOTES

Size: px
Start display at page:

Download "COMBINATORIAL GROUP THEORY NOTES"

Transcription

1 COMBINATORIAL GROUP THEORY NOTES These are being written as a companion to Chapter 1 of Hatcher. The aim is to give a description of some of the group theory required to work with the fundamental groups of the spaces we have been studying. The adjective combinatorial is used, roughly speaking, to describe the viewpoint of studying groups via words in a generating set of a group. It s also used to distinguish this approach from more modern geometric and topological approaches to studying finitely generated groups. In this course we will mostly be using group theory as a tool for studying topological spaces, but it is worth noting that in general the flow of information between group theory and topology can travel in both directions. The central tool for understanding both free products of groups and presentations of groups is the notion of a word: Definition 0.1. A word on a set S is a finite, ordered sequence of elements of S, which we write as s 1, s 2,..., s k or sometimes s 1 s 2 s k. We allow this list to be empty, in which case we call this the empty word, and write the set of all words in S as W (S). The set S is sometimes called the alphabet over which we are choosing words, with each element of S as a letter in this alphabet. There is a natural product structure on W (S) given by concatenation of words. The alphabets that we shall study come in two flavours, the first being an alphabet S = α I G α which contains all elements from a family of groups. The free product of this family is then defined by an equivalence relation over the set of all words in S. The second type of alphabet is of the form X X 1, where X is a generating set of a group G. A presentation X R of G gives a way to describe the group in terms of an equivalence relation on W (X X 1 ), or equivalently, as a quotient of the free group on the set X. This will be studied in the second section. 1. Free products of groups For this section we shall fix a set (G α ) α I of groups indexed by a set I, and we fix S = α I G α to be the alphabet consisting of the union of all elements in these groups. Definition 1.1. With S defined as above, let w = s 1 s k be a word in S, and suppose that G αi is the group in our above set which contains 1

2 2 COMBINATORIAL GROUP THEORY NOTES s i. We define four elementary moves on a word w W (S). There are two down moves: (D1) Remove a letter s i if s i = 1 in G αi. (D2) Replace the subword s i, s i+1 with the letter s i s i+1 if both letters belong to the same group (i.e. α i = α i+1.) Also, we define two up moves (U1) and (U2) to be the reverse operations of (D1) and (D2). The elementary moves give an equivalence relation on the set of words in S, and we let [w] denote the equivalence class of each word. If w 1 w 1 and w 2 w 2 then w 1.w 2 w 1.w 2, therefore concatenation of words descends to a product operation on the set of equivalence classes. Definition 1.2. The free product α I G α is defined to be the set of equivalence classes of words in W (S)/. Multiplication in α I G α is given by [w].[w ] = [w.w ], and if w = s 1,..., s k then the inverse of [w] is given by the equivalence class of the word s 1 k, s 1 k 1,..., s 1 1. The identity element is the equivalence class of the empty word in W (S). Despite having a perfectly satisfactory definition of the free product, this does not give us a way of distinguishing between two elements. For that we need the notion of reduced words: Definition 1.3. A word w = s 1, s 2,..., s k in a free product α I G α is reduced if no moves of the form (D1) or (D2) can be performed on w. Given any word in w W (S), one can obtain a reduced word w from w via applying a sequence of elementary moves of the form (D1) and (D2) until no such moves are possible. This process eventually terminates as applying a down move reduces the length of a word. The main theorem which allows us to work with free products is the following: Theorem 1.4. Each equivalence class [w] contains a unique reduced word. Proof. Suppose that w w and let w = w 1, w 2, w 3,..., w n = w be a sequence of words in W (S) such that each w i+1 is obtained from w i via an elementary move. We shall show that we can alter this sequence to obtain a new sequence of elementary moves between w and w such that we perform all the down moves first, followed by all the up moves second. This is enough to prove the theorem: if w and w are both reduced then no down moves can be performed on w, and

3 COMBINATORIAL GROUP THEORY NOTES 3 w cannot be preceded by an up move. It follows that this altered sequence must be trivial, and w = w. We work left to right in the sequence, and need to edit pieces of the form w i 1 (Ua) w i (Db) w i+1, with a, b {1, 2}. One can show that either w i 1 = w i+1 (in which case we can remove w i from the sequence entirely) or there exists w i and a down and an up move such that w i 1 w i w i+1. The proof is case by case, and we shall omit the details. As an example, if both a = b = 1 then w i is obtained from w i 1 by adding a letter equal to the identity in a group, and w i+1 is obtained from w i by removing a letter equal to an identity element. Either the added and removed letters are identical, in which case w i 1 = w i+1 and we can omit both these moves, or the moves occur in different parts of the word w i and so commute. One can remove the identity element for the down move first to obtain a word w i before adding the identity element given by the up move to get w i+1. This theorem has several useful consequences. Firstly, it tells us that if a reduced word is non-empty then it represents a nontrivial element of the free product. Uniqueness of reduced words also tells us how to decide if two equivalence classes [w] and [w ] are equal: find the reduced representatives in each equivalence class and compare them letter by letter. In particular, words containing a single letter are nontrivial, and this implies that for each G β there is an injective homomorphism ι β : G β α I G α given by taking an element g G β to the word in the free product consisting of the single letter g. We will blur this distinction, and view G β as a subgroup of α I G α without referring explicitly to this injection. These subgroups control homomorphisms from the free product to another group H in a very precise way: Theorem 1.5 (The Universal Property of Free Products). A set of homomorphisms {φ α : G α H} induces a unique homomorphism φ : α I G α H such that φ Gα = φ α for all α I. Proof. Suppose we are given such a set (φ α ) of homomorphisms. Given w = s 1,..., s k with s i G αi, define φ(w) = φ α1 (s 1 )φ α2 (s 2 ) φ αk (s k ) One can check that changing w by an elementary move does not change the image of w in H, and this descends to a homomorphism from the free product to H such that φ Gα = φ α. As φ is a homomorphism and

4 4 COMBINATORIAL GROUP THEORY NOTES multiplication in the free product is given by concatenation, we only need to know the values of φ on each subgroup G α in order to determine φ on the whole of the free product. Hence such a homomorphism φ is unique. 2. Free groups For this section we consider words on a set X X 1. The set X 1 is considered as disjoint from X and in bijection with X via a (formal) inversion map. The approach to this section is similar to the one for free products, however as free groups are easier to work with, it is worth stating the results separately in a slightly more convenient fashion. Definition 2.1. An elementary move on a word w = s 1, s 2,..., s k W (X X 1 ) changes w in one of two ways: (D1): Remove the subword s i, s i+1 if s i = s 1 i+1. (U1): Insert a subword x 1, x into w for some x X X 1 We define an equivalence relation on W (X X 1 ) by saying that w w if w can be obtained from w by a sequence of elementary moves. Similar to the free product situation, this equivalence relation respects concatenation of words, so we may define a product operation on equivalence classes by [w].[w ] = [w.w ] Definition 2.2. The free group F (X) on the set X is defined to be the set of equivalence classes of words in W (X X 1 ) with the product operation. The empty word is the identity in F (X) and inversion is given by taking the equivalence class of the word s 1, s 2,..., s k to the equivalence class of the word s 1 k, s 1 k 1,..., s 1 1. This is equivalent to the definition of free group given in class. It is an exercise to show that F (X) is isomorphic to a free product of X copies of Z. Definition 2.3. A word s 1,..., s k in W (X X 1 ) is said to be reduced if s i s 1 i+1 for all i. Example 2.4. It is convenient to collect terms together, so that rather than write x, x, y, y, y as an element of F ({x, y}), we instead write x 2 y 3. In this sense, the word x 2 y 3 is reduced, however the word xx 1 yx is not. As with the free product case, given any word w we can find a reduced word w in the equivalence class of w by applying a sequence of (D1) moves to w. We also have the same useful proposition:

5 COMBINATORIAL GROUP THEORY NOTES 5 Proposition 2.5. A reduced word w obtained from w by a sequence of elementary reductions is independent of the choice of reduction sequence. This is proved in a similar manner to Theorem 1.4. Thus we know that two elements of F (X) are equivalent if and only their reduced forms are equivalent. As a result of this we will often blur the distinction between an element of F (X) and a word in its equivalence class. We also have a universal property for free groups, which we can phrase in a slightly more convenient form: Proposition 2.6 (The Universal Property of Free Groups). Any map f : X G from X to a group G extends uniquely to a homomorphism φ : F (X) G. In other words, we only need to choose images of the elements of X to define a homomorphism from F (X) to a group G, and any homomorphism from a free group is determined by the images of the generators. 3. Generating sets and normal generating sets Given any subset X G, there is a smallest subgroup of G which contains X it is the intersection of all subgroups of G containing X. We write this group as X and say that this is the subgroup of G generated by X. Any element in X can be written a product of elements in X, and X can alternatively be viewed as the image of the homomorphism φ : F (X) G induced by the universal property of free groups and the natural injection f : X G. Definition 3.1. We say that G is finitely generated if there exists a finite set X such that X = G. There is also a smallest normal subgroup of G containing X given by the intersection of all normal subgroups of G which contain X. We write this as X, and say this is the subgroup of G normally generated by the set X. Example 3.2. Let R be a subset of F (X). The normal subgroup R consists of equivalence classes of words of the form k w i r ɛ i i w 1 i, i=1 where each w i W (X X 1 ), each r i is a representative of an element of R, and each ɛ i = ±1.

6 6 COMBINATORIAL GROUP THEORY NOTES Definition 3.3. A finitely presented group is a group of the form G = F (X)/ R where X is a finite set and R is a finite subset of F (X). We write this as G = X R. If we already have an abstract group G, then a finite presentation of G is given by a finite generating set X of G, along with a finite set R which normally generates the kernel of the natural map φ : F (X) G (note that by the first isomorphism theorem this implies that G is isomorphic to F (X)/ R. Suppose that G is finitely presented, so that we can associate G with F (X)/ R. In this case two elements w, w of F (X) have the same image in G if and only if they lie in the same coset of R. In other words (1) w w. k i=1 w i r ɛ i i w 1 i, where the product is a word as described in the above example, and implies both sides have the same reduced representatives. The set R can be seen as giving an equivalence relation R on the set of words in X X 1 where w R w if and only if w and w satisfy the an equation of the form (1). Example 3.4. where [x, y] = xyx 1 y 1 Z Z = x, y [x, y] Proof. By the universal property of free groups there exists a homomorphism φ : F (x, y) Z Z induced by mapping x to (1, 0) and y to (0, 1). This is clearly surjective, so to prove that the above does indeed give a presentation of Z Z, we need to show that ker φ is normally generated by [x, y]. As [x, y] ker φ it follows that the normal subgroup generated by [x, y] lies in ker φ. Conversely, suppose that w is a word in the kernel of φ. Then w has a reduced representative of the form w = x k 1 y l 1 x k 2 y l2 x kn y ln, where k i = l i = 0. The computation xy = yx.(x 1 y 1 )[x, y](x 1 y 1 ) 1 tells us that xy R yx, so we can swap occurrences of x and y and remain in the same equivalence class in R, to tell us that x k 1 y l 1 x k 2 y l2 x kn y ln R x k i y l i = x 0 y 0 = 1 Hence w R and ker φ = R.

7 COMBINATORIAL GROUP THEORY NOTES 7 4. Abelianizing presentations With some presentations, such as that for a free group and the abelian group above, it is easy to decide if two words represent the same element of the underlying group. However, this is far from possible in general: Theorem 4.1 (Novikov, 1955). There does not exist an algorithm to decide whether an arbitrary finitely presented group X R is nontrivial. Of course, the above theorem is frustrating when we are spending time trying to distinguish between topological spaces via their fundamental groups. Luckily, the presentations we will be studying will be easier to get to grips with than the examples given by Novikov. We can often pass to the abelianization of a group, which is something that can be determined by a presentation. Definition 4.2. The commutator subgroup of a group is defined by: [G, G] = {[g, h] : g G, h H} It is the subgroup generated by commutators of elements in G (elements of the form [g, h] = ghg 1 h 1 ). The subgroup [G, G] is a normal subgroup of G (exercise!), and we define the abelianization of G to be the quotient G ab = G/[G, G] The abelianization of a group G is the largest abelian quotient of G. This is made precise by the following proposition: Proposition 4.3. The quotient G ab is abelian. If φ : G H is any homomorphism from G to an abelian group H then φ descends to a homomorphism φ : G ab H which factors through the quotient map q : G G ab. Unlike the group itself, the abelianization of a finitely presented group is algorithmically decideable. We will give a sketch. Suppose that G = X R is a finitely presented group. Write X = {x 1,..., x n } and let A = Z[X] be the direct sum of n copies of Z, where each coordinate in a tuple (k 1, k 2,..., k n ) A corresponds to an element of X. Each r R gives an element r A where each co-ordinate is given by summing the indices of the corresponding element of X occurring in r. Then R = r : r R is a (normal) subgroup of A, and the abelianization of G is isomorphic to A/R. This is something that can be computed relatively easily

8 8 COMBINATORIAL GROUP THEORY NOTES (think vector spaces exercises to follow). In particular, as the standard presentation X of the free group on X has no relations, its abelianization is isomorphic to A.

BASIC GROUP THEORY : G G G,

BASIC GROUP THEORY : G G G, BASIC GROUP THEORY 18.904 1. Definitions Definition 1.1. A group (G, ) is a set G with a binary operation : G G G, and a unit e G, possessing the following properties. (1) Unital: for g G, we have g e

More information

Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra

Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................

More information

WHY WORD PROBLEMS ARE HARD

WHY WORD PROBLEMS ARE HARD WHY WORD PROBLEMS ARE HARD KEITH CONRAD 1. Introduction The title above is a joke. Many students in school hate word problems. We will discuss here a specific math question that happens to be named the

More information

Teddy Einstein Math 4320

Teddy Einstein Math 4320 Teddy Einstein Math 4320 HW4 Solutions Problem 1: 2.92 An automorphism of a group G is an isomorphism G G. i. Prove that Aut G is a group under composition. Proof. Let f, g Aut G. Then f g is a bijective

More information

ABSTRACT ALGEBRA 1, LECTURES NOTES 5: SUBGROUPS, CONJUGACY, NORMALITY, QUOTIENT GROUPS, AND EXTENSIONS.

ABSTRACT ALGEBRA 1, LECTURES NOTES 5: SUBGROUPS, CONJUGACY, NORMALITY, QUOTIENT GROUPS, AND EXTENSIONS. ABSTRACT ALGEBRA 1, LECTURES NOTES 5: SUBGROUPS, CONJUGACY, NORMALITY, QUOTIENT GROUPS, AND EXTENSIONS. ANDREW SALCH 1. Subgroups, conjugacy, normality. I think you already know what a subgroup is: Definition

More information

CS 468: Computational Topology Group Theory Fall b c b a b a c b a c b c c b a

CS 468: Computational Topology Group Theory Fall b c b a b a c b a c b c c b a Q: What s purple and commutes? A: An abelian grape! Anonymous Group Theory Last lecture, we learned about a combinatorial method for characterizing spaces: using simplicial complexes as triangulations

More information

Lectures - XXIII and XXIV Coproducts and Pushouts

Lectures - XXIII and XXIV Coproducts and Pushouts Lectures - XXIII and XXIV Coproducts and Pushouts We now discuss further categorical constructions that are essential for the formulation of the Seifert Van Kampen theorem. We first discuss the notion

More information

Stab(t) = {h G h t = t} = {h G h (g s) = g s} = {h G (g 1 hg) s = s} = g{k G k s = s} g 1 = g Stab(s)g 1.

Stab(t) = {h G h t = t} = {h G h (g s) = g s} = {h G (g 1 hg) s = s} = g{k G k s = s} g 1 = g Stab(s)g 1. 1. Group Theory II In this section we consider groups operating on sets. This is not particularly new. For example, the permutation group S n acts on the subset N n = {1, 2,...,n} of N. Also the group

More information

THE FUNDAMENTAL GROUP AND SEIFERT-VAN KAMPEN S THEOREM

THE FUNDAMENTAL GROUP AND SEIFERT-VAN KAMPEN S THEOREM THE FUNDAMENTAL GROUP AND SEIFERT-VAN KAMPEN S THEOREM KATHERINE GALLAGHER Abstract. The fundamental group is an essential tool for studying a topological space since it provides us with information about

More information

Equational Logic. Chapter Syntax Terms and Term Algebras

Equational Logic. Chapter Syntax Terms and Term Algebras Chapter 2 Equational Logic 2.1 Syntax 2.1.1 Terms and Term Algebras The natural logic of algebra is equational logic, whose propositions are universally quantified identities between terms built up from

More information

MATH 101: ALGEBRA I WORKSHEET, DAY #3. Fill in the blanks as we finish our first pass on prerequisites of group theory.

MATH 101: ALGEBRA I WORKSHEET, DAY #3. Fill in the blanks as we finish our first pass on prerequisites of group theory. MATH 101: ALGEBRA I WORKSHEET, DAY #3 Fill in the blanks as we finish our first pass on prerequisites of group theory 1 Subgroups, cosets Let G be a group Recall that a subgroup H G is a subset that is

More information

De Rham Cohomology. Smooth singular cochains. (Hatcher, 2.1)

De Rham Cohomology. Smooth singular cochains. (Hatcher, 2.1) II. De Rham Cohomology There is an obvious similarity between the condition d o q 1 d q = 0 for the differentials in a singular chain complex and the condition d[q] o d[q 1] = 0 which is satisfied by the

More information

ALGEBRA II: RINGS AND MODULES OVER LITTLE RINGS.

ALGEBRA II: RINGS AND MODULES OVER LITTLE RINGS. ALGEBRA II: RINGS AND MODULES OVER LITTLE RINGS. KEVIN MCGERTY. 1. RINGS The central characters of this course are algebraic objects known as rings. A ring is any mathematical structure where you can add

More information

Math 6510 Homework 11

Math 6510 Homework 11 2.2 Problems 40 Problem. From the long exact sequence of homology groups associted to the short exact sequence of chain complexes n 0 C i (X) C i (X) C i (X; Z n ) 0, deduce immediately that there are

More information

2 Lecture 2: Logical statements and proof by contradiction Lecture 10: More on Permutations, Group Homomorphisms 31

2 Lecture 2: Logical statements and proof by contradiction Lecture 10: More on Permutations, Group Homomorphisms 31 Contents 1 Lecture 1: Introduction 2 2 Lecture 2: Logical statements and proof by contradiction 7 3 Lecture 3: Induction and Well-Ordering Principle 11 4 Lecture 4: Definition of a Group and examples 15

More information

The Hurewicz Theorem

The Hurewicz Theorem The Hurewicz Theorem April 5, 011 1 Introduction The fundamental group and homology groups both give extremely useful information, particularly about path-connected spaces. Both can be considered as functors,

More information

THE FUNDAMENTAL GROUP AND CW COMPLEXES

THE FUNDAMENTAL GROUP AND CW COMPLEXES THE FUNDAMENTAL GROUP AND CW COMPLEXES JAE HYUNG SIM Abstract. This paper is a quick introduction to some basic concepts in Algebraic Topology. We start by defining homotopy and delving into the Fundamental

More information

Winter School on Galois Theory Luxembourg, February INTRODUCTION TO PROFINITE GROUPS Luis Ribes Carleton University, Ottawa, Canada

Winter School on Galois Theory Luxembourg, February INTRODUCTION TO PROFINITE GROUPS Luis Ribes Carleton University, Ottawa, Canada Winter School on alois Theory Luxembourg, 15-24 February 2012 INTRODUCTION TO PROFINITE ROUPS Luis Ribes Carleton University, Ottawa, Canada LECTURE 2 2.1 ENERATORS OF A PROFINITE ROUP 2.2 FREE PRO-C ROUPS

More information

Homework 3 MTH 869 Algebraic Topology

Homework 3 MTH 869 Algebraic Topology Homework 3 MTH 869 Algebraic Topology Joshua Ruiter February 12, 2018 Proposition 0.1 (Exercise 1.1.10). Let (X, x 0 ) and (Y, y 0 ) be pointed, path-connected spaces. Let f : I X y 0 } and g : I x 0 }

More information

Corrections to Introduction to Topological Manifolds (First edition) by John M. Lee December 7, 2015

Corrections to Introduction to Topological Manifolds (First edition) by John M. Lee December 7, 2015 Corrections to Introduction to Topological Manifolds (First edition) by John M. Lee December 7, 2015 Changes or additions made in the past twelve months are dated. Page 29, statement of Lemma 2.11: The

More information

A Primer on Homological Algebra

A Primer on Homological Algebra A Primer on Homological Algebra Henry Y Chan July 12, 213 1 Modules For people who have taken the algebra sequence, you can pretty much skip the first section Before telling you what a module is, you probably

More information

CHAPTER III NORMAL SERIES

CHAPTER III NORMAL SERIES CHAPTER III NORMAL SERIES 1. Normal Series A group is called simple if it has no nontrivial, proper, normal subgroups. The only abelian simple groups are cyclic groups of prime order, but some authors

More information

LECTURE 3: RELATIVE SINGULAR HOMOLOGY

LECTURE 3: RELATIVE SINGULAR HOMOLOGY LECTURE 3: RELATIVE SINGULAR HOMOLOGY In this lecture we want to cover some basic concepts from homological algebra. These prove to be very helpful in our discussion of singular homology. The following

More information

Notes on Geometry of Surfaces

Notes on Geometry of Surfaces Notes on Geometry of Surfaces Contents Chapter 1. Fundamental groups of Graphs 5 1. Review of group theory 5 2. Free groups and Ping-Pong Lemma 8 3. Subgroups of free groups 15 4. Fundamental groups of

More information

NOTES ON AUTOMATA. Date: April 29,

NOTES ON AUTOMATA. Date: April 29, NOTES ON AUTOMATA 1. Monoids acting on sets We say that a monoid S with identity element ɛ acts on a set Q if q(st) = (qs)t and qɛ = q. As with groups, if we set s = t whenever qs = qt for all q Q, then

More information

Elements of solution for Homework 5

Elements of solution for Homework 5 Elements of solution for Homework 5 General remarks How to use the First Isomorphism Theorem A standard way to prove statements of the form G/H is isomorphic to Γ is to construct a homomorphism ϕ : G Γ

More information

Math 637 Topology Paulo Lima-Filho. Problem List I. b. Show that a contractible space is path connected.

Math 637 Topology Paulo Lima-Filho. Problem List I. b. Show that a contractible space is path connected. Problem List I Problem 1. A space X is said to be contractible if the identiy map i X : X X is nullhomotopic. a. Show that any convex subset of R n is contractible. b. Show that a contractible space is

More information

1 Differentiable manifolds and smooth maps

1 Differentiable manifolds and smooth maps 1 Differentiable manifolds and smooth maps Last updated: April 14, 2011. 1.1 Examples and definitions Roughly, manifolds are sets where one can introduce coordinates. An n-dimensional manifold is a set

More information

Definitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations

Definitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of

More information

Normal forms in combinatorial algebra

Normal forms in combinatorial algebra Alberto Gioia Normal forms in combinatorial algebra Master s thesis, defended on July 8, 2009 Thesis advisor: Hendrik Lenstra Mathematisch Instituut Universiteit Leiden ii Contents Introduction iv 1 Generators

More information

Basic Concepts of Group Theory

Basic Concepts of Group Theory Chapter 1 Basic Concepts of Group Theory The theory of groups and vector spaces has many important applications in a number of branches of modern theoretical physics. These include the formal theory of

More information

UNIVERSAL DERIVED EQUIVALENCES OF POSETS

UNIVERSAL DERIVED EQUIVALENCES OF POSETS UNIVERSAL DERIVED EQUIVALENCES OF POSETS SEFI LADKANI Abstract. By using only combinatorial data on two posets X and Y, we construct a set of so-called formulas. A formula produces simultaneously, for

More information

CONSEQUENCES OF THE SYLOW THEOREMS

CONSEQUENCES OF THE SYLOW THEOREMS CONSEQUENCES OF THE SYLOW THEOREMS KEITH CONRAD For a group theorist, Sylow s Theorem is such a basic tool, and so fundamental, that it is used almost without thinking, like breathing. Geoff Robinson 1.

More information

FUNDAMENTAL GROUPS AND THE VAN KAMPEN S THEOREM. Contents

FUNDAMENTAL GROUPS AND THE VAN KAMPEN S THEOREM. Contents FUNDAMENTAL GROUPS AND THE VAN KAMPEN S THEOREM SAMUEL BLOOM Abstract. In this paper, we define the fundamental group of a topological space and explore its structure, and we proceed to prove Van-Kampen

More information

An Algebraic View of the Relation between Largest Common Subtrees and Smallest Common Supertrees

An Algebraic View of the Relation between Largest Common Subtrees and Smallest Common Supertrees An Algebraic View of the Relation between Largest Common Subtrees and Smallest Common Supertrees Francesc Rosselló 1, Gabriel Valiente 2 1 Department of Mathematics and Computer Science, Research Institute

More information

Rohit Garg Roll no Dr. Deepak Gumber

Rohit Garg Roll no Dr. Deepak Gumber FINITE -GROUPS IN WHICH EACH CENTRAL AUTOMORPHISM FIXES THE CENTER ELEMENTWISE Thesis submitted in partial fulfillment of the requirement for the award of the degree of Masters of Science In Mathematics

More information

Groups and Symmetries

Groups and Symmetries Groups and Symmetries Definition: Symmetry A symmetry of a shape is a rigid motion that takes vertices to vertices, edges to edges. Note: A rigid motion preserves angles and distances. Definition: Group

More information

2. Intersection Multiplicities

2. Intersection Multiplicities 2. Intersection Multiplicities 11 2. Intersection Multiplicities Let us start our study of curves by introducing the concept of intersection multiplicity, which will be central throughout these notes.

More information

Solutions to Problem Set 1

Solutions to Problem Set 1 Solutions to Problem Set 1 18.904 Spring 2011 Problem 1 Statement. Let n 1 be an integer. Let CP n denote the set of all lines in C n+1 passing through the origin. There is a natural map π : C n+1 \ {0}

More information

1.5 Applications Of The Sylow Theorems

1.5 Applications Of The Sylow Theorems 14 CHAPTER1. GROUP THEORY 8. The Sylow theorems are about subgroups whose order is a power of a prime p. Here is a result about subgroups of index p. Let H be a subgroup of the finite group G, and assume

More information

6 More on simple groups Lecture 20: Group actions and simplicity Lecture 21: Simplicity of some group actions...

6 More on simple groups Lecture 20: Group actions and simplicity Lecture 21: Simplicity of some group actions... 510A Lecture Notes 2 Contents I Group theory 5 1 Groups 7 1.1 Lecture 1: Basic notions............................................... 8 1.2 Lecture 2: Symmetries and group actions......................................

More information

Solutions to Assignment 4

Solutions to Assignment 4 1. Let G be a finite, abelian group written additively. Let x = g G g, and let G 2 be the subgroup of G defined by G 2 = {g G 2g = 0}. (a) Show that x = g G 2 g. (b) Show that x = 0 if G 2 = 2. If G 2

More information

A Little Beyond: Linear Algebra

A Little Beyond: Linear Algebra A Little Beyond: Linear Algebra Akshay Tiwary March 6, 2016 Any suggestions, questions and remarks are welcome! 1 A little extra Linear Algebra 1. Show that any set of non-zero polynomials in [x], no two

More information

HOMOLOGY AND COHOMOLOGY. 1. Introduction

HOMOLOGY AND COHOMOLOGY. 1. Introduction HOMOLOGY AND COHOMOLOGY ELLEARD FELIX WEBSTER HEFFERN 1. Introduction We have been introduced to the idea of homology, which derives from a chain complex of singular or simplicial chain groups together

More information

FREE PRODUCTS AND BRITTON S LEMMA

FREE PRODUCTS AND BRITTON S LEMMA FREE PRODUCTS AND BRITTON S LEMMA Dan Lidral-Porter 1. Free Products I determined that the best jumping off point was to start with free products. Free products are an extension of the notion of free groups.

More information

Math 145. Codimension

Math 145. Codimension Math 145. Codimension 1. Main result and some interesting examples In class we have seen that the dimension theory of an affine variety (irreducible!) is linked to the structure of the function field in

More information

Category Theory. Categories. Definition.

Category Theory. Categories. Definition. Category Theory Category theory is a general mathematical theory of structures, systems of structures and relationships between systems of structures. It provides a unifying and economic mathematical modeling

More information

Monoids. Definition: A binary operation on a set M is a function : M M M. Examples:

Monoids. Definition: A binary operation on a set M is a function : M M M. Examples: Monoids Definition: A binary operation on a set M is a function : M M M. If : M M M, we say that is well defined on M or equivalently, that M is closed under the operation. Examples: Definition: A monoid

More information

7.3 Singular Homology Groups

7.3 Singular Homology Groups 184 CHAPTER 7. HOMOLOGY THEORY 7.3 Singular Homology Groups 7.3.1 Cycles, Boundaries and Homology Groups We can define the singular p-chains with coefficients in a field K. Furthermore, we can define the

More information

chapter 12 MORE MATRIX ALGEBRA 12.1 Systems of Linear Equations GOALS

chapter 12 MORE MATRIX ALGEBRA 12.1 Systems of Linear Equations GOALS chapter MORE MATRIX ALGEBRA GOALS In Chapter we studied matrix operations and the algebra of sets and logic. We also made note of the strong resemblance of matrix algebra to elementary algebra. The reader

More information

Algebraic Topology. Oscar Randal-Williams. or257/teaching/notes/at.pdf

Algebraic Topology. Oscar Randal-Williams.   or257/teaching/notes/at.pdf Algebraic Topology Oscar Randal-Williams https://www.dpmms.cam.ac.uk/ or257/teaching/notes/at.pdf 1 Introduction 1 1.1 Some recollections and conventions...................... 2 1.2 Cell complexes.................................

More information

Equivalence Relations and Partitions, Normal Subgroups, Quotient Groups, and Homomorphisms

Equivalence Relations and Partitions, Normal Subgroups, Quotient Groups, and Homomorphisms Equivalence Relations and Partitions, Normal Subgroups, Quotient Groups, and Homomorphisms Math 356 Abstract We sum up the main features of our last three class sessions, which list of topics are given

More information

Written Homework # 2 Solution

Written Homework # 2 Solution Math 516 Fall 2006 Radford Written Homework # 2 Solution 10/09/06 Let G be a non-empty set with binary operation. For non-empty subsets S, T G we define the product of the sets S and T by If S = {s} is

More information

INVERSE LIMITS AND PROFINITE GROUPS

INVERSE LIMITS AND PROFINITE GROUPS INVERSE LIMITS AND PROFINITE GROUPS BRIAN OSSERMAN We discuss the inverse limit construction, and consider the special case of inverse limits of finite groups, which should best be considered as topological

More information

Notes on Group Theory. by Avinash Sathaye, Professor of Mathematics November 5, 2013

Notes on Group Theory. by Avinash Sathaye, Professor of Mathematics November 5, 2013 Notes on Group Theory by Avinash Sathaye, Professor of Mathematics November 5, 2013 Contents 1 Preparation. 2 2 Group axioms and definitions. 2 Shortcuts................................. 2 2.1 Cyclic groups............................

More information

Course 311: Abstract Algebra Academic year

Course 311: Abstract Algebra Academic year Course 311: Abstract Algebra Academic year 2007-08 D. R. Wilkins Copyright c David R. Wilkins 1997 2007 Contents 1 Topics in Group Theory 1 1.1 Groups............................... 1 1.2 Examples of Groups.......................

More information

2 Permutation Groups

2 Permutation Groups 2 Permutation Groups Last Time Orbit/Stabilizer algorithm: Orbit of a point. Transversal of transporter elements. Generators for stabilizer. Today: Use in a ``divide-and-conquer approach for permutation

More information

Universal Algebra for Logics

Universal Algebra for Logics Universal Algebra for Logics Joanna GRYGIEL University of Czestochowa Poland j.grygiel@ajd.czest.pl 2005 These notes form Lecture Notes of a short course which I will give at 1st School on Universal Logic

More information

Automata on linear orderings

Automata on linear orderings Automata on linear orderings Véronique Bruyère Institut d Informatique Université de Mons-Hainaut Olivier Carton LIAFA Université Paris 7 September 25, 2006 Abstract We consider words indexed by linear

More information

Notes 2 for MAT4270 Connected components and universal covers π 0 and π 1.

Notes 2 for MAT4270 Connected components and universal covers π 0 and π 1. Notes 2 for MAT4270 Connected components and universal covers π 0 and π 1. Version 0.00 with misprints, Connected components Recall thaty if X is a topological space X is said to be connected if is not

More information

Groups of Prime Power Order with Derived Subgroup of Prime Order

Groups of Prime Power Order with Derived Subgroup of Prime Order Journal of Algebra 219, 625 657 (1999) Article ID jabr.1998.7909, available online at http://www.idealibrary.com on Groups of Prime Power Order with Derived Subgroup of Prime Order Simon R. Blackburn*

More information

Rings and Fields Theorems

Rings and Fields Theorems Rings and Fields Theorems Rajesh Kumar PMATH 334 Intro to Rings and Fields Fall 2009 October 25, 2009 12 Rings and Fields 12.1 Definition Groups and Abelian Groups Let R be a non-empty set. Let + and (multiplication)

More information

Tree-adjoined spaces and the Hawaiian earring

Tree-adjoined spaces and the Hawaiian earring Tree-adjoined spaces and the Hawaiian earring W. Hojka (TU Wien) Workshop on Fractals and Tilings 2009 July 6-10, 2009, Strobl (Austria) W. Hojka (TU Wien) () Tree-adjoined spaces and the Hawaiian earring

More information

Math 4400, Spring 08, Sample problems Final Exam.

Math 4400, Spring 08, Sample problems Final Exam. Math 4400, Spring 08, Sample problems Final Exam. 1. Groups (1) (a) Let a be an element of a group G. Define the notions of exponent of a and period of a. (b) Suppose a has a finite period. Prove that

More information

NOTES IN COMMUTATIVE ALGEBRA: PART 2

NOTES IN COMMUTATIVE ALGEBRA: PART 2 NOTES IN COMMUTATIVE ALGEBRA: PART 2 KELLER VANDEBOGERT 1. Completion of a Ring/Module Here we shall consider two seemingly different constructions for the completion of a module and show that indeed they

More information

8 Complete fields and valuation rings

8 Complete fields and valuation rings 18.785 Number theory I Fall 2017 Lecture #8 10/02/2017 8 Complete fields and valuation rings In order to make further progress in our investigation of finite extensions L/K of the fraction field K of a

More information

ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ.

ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ. ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ. ANDREW SALCH 1. Hilbert s Nullstellensatz. The last lecture left off with the claim that, if J k[x 1,..., x n ] is an ideal, then

More information

HOMOLOGY THEORIES INGRID STARKEY

HOMOLOGY THEORIES INGRID STARKEY HOMOLOGY THEORIES INGRID STARKEY Abstract. This paper will introduce the notion of homology for topological spaces and discuss its intuitive meaning. It will also describe a general method that is used

More information

Algebraic Geometry. Andreas Gathmann. Class Notes TU Kaiserslautern 2014

Algebraic Geometry. Andreas Gathmann. Class Notes TU Kaiserslautern 2014 Algebraic Geometry Andreas Gathmann Class Notes TU Kaiserslautern 2014 Contents 0. Introduction......................... 3 1. Affine Varieties........................ 9 2. The Zariski Topology......................

More information

INTRODUCTION TO SEMIGROUPS AND MONOIDS

INTRODUCTION TO SEMIGROUPS AND MONOIDS INTRODUCTION TO SEMIGROUPS AND MONOIDS PETE L. CLARK We give here some basic definitions and very basic results concerning semigroups and monoids. Aside from the mathematical maturity necessary to follow

More information

2. Prime and Maximal Ideals

2. Prime and Maximal Ideals 18 Andreas Gathmann 2. Prime and Maximal Ideals There are two special kinds of ideals that are of particular importance, both algebraically and geometrically: the so-called prime and maximal ideals. Let

More information

Math 121 Homework 5: Notes on Selected Problems

Math 121 Homework 5: Notes on Selected Problems Math 121 Homework 5: Notes on Selected Problems 12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x 1,..., x n is any maximal set of linearly independent elements

More information

Math 530 Lecture Notes. Xi Chen

Math 530 Lecture Notes. Xi Chen Math 530 Lecture Notes Xi Chen 632 Central Academic Building, University of Alberta, Edmonton, Alberta T6G 2G1, CANADA E-mail address: xichen@math.ualberta.ca 1991 Mathematics Subject Classification. Primary

More information

(dim Z j dim Z j 1 ) 1 j i

(dim Z j dim Z j 1 ) 1 j i Math 210B. Codimension 1. Main result and some interesting examples Let k be a field, and A a domain finitely generated k-algebra. In class we have seen that the dimension theory of A is linked to the

More information

3. The Sheaf of Regular Functions

3. The Sheaf of Regular Functions 24 Andreas Gathmann 3. The Sheaf of Regular Functions After having defined affine varieties, our next goal must be to say what kind of maps between them we want to consider as morphisms, i. e. as nice

More information

ABSTRACT ALGEBRA 1, LECTURE NOTES 5: HOMOMORPHISMS, ISOMORPHISMS, SUBGROUPS, QUOTIENT ( FACTOR ) GROUPS. ANDREW SALCH

ABSTRACT ALGEBRA 1, LECTURE NOTES 5: HOMOMORPHISMS, ISOMORPHISMS, SUBGROUPS, QUOTIENT ( FACTOR ) GROUPS. ANDREW SALCH ABSTRACT ALGEBRA 1, LECTURE NOTES 5: HOMOMORPHISMS, ISOMORPHISMS, SUBGROUPS, QUOTIENT ( FACTOR ) GROUPS. ANDREW SALCH 1. Homomorphisms and isomorphisms between groups. Definition 1.1. Let G, H be groups.

More information

Math 249B. Geometric Bruhat decomposition

Math 249B. Geometric Bruhat decomposition Math 249B. Geometric Bruhat decomposition 1. Introduction Let (G, T ) be a split connected reductive group over a field k, and Φ = Φ(G, T ). Fix a positive system of roots Φ Φ, and let B be the unique

More information

Tree sets. Reinhard Diestel

Tree sets. Reinhard Diestel 1 Tree sets Reinhard Diestel Abstract We study an abstract notion of tree structure which generalizes treedecompositions of graphs and matroids. Unlike tree-decompositions, which are too closely linked

More information

Commutative Algebra MAS439 Lecture 3: Subrings

Commutative Algebra MAS439 Lecture 3: Subrings Commutative Algebra MAS439 Lecture 3: Subrings Paul Johnson paul.johnson@sheffield.ac.uk Hicks J06b October 4th Plan: slow down a little Last week - Didn t finish Course policies + philosophy Sections

More information

CHAPTER 3: THE INTEGERS Z

CHAPTER 3: THE INTEGERS Z CHAPTER 3: THE INTEGERS Z MATH 378, CSUSM. SPRING 2009. AITKEN 1. Introduction The natural numbers are designed for measuring the size of finite sets, but what if you want to compare the sizes of two sets?

More information

n ) = f (x 1 ) e 1... f (x n ) e n

n ) = f (x 1 ) e 1... f (x n ) e n 1. FREE GROUPS AND PRESENTATIONS Let X be a subset of a group G. The subgroup generated by X, denoted X, is the intersection of all subgroups of G containing X as a subset. If g G, then g X g can be written

More information

1. Quivers and their representations: Basic definitions and examples.

1. Quivers and their representations: Basic definitions and examples. 1 Quivers and their representations: Basic definitions and examples 11 Quivers A quiver Q (sometimes also called a directed graph) consists of vertices and oriented edges (arrows): loops and multiple arrows

More information

Vector Spaces. Chapter 1

Vector Spaces. Chapter 1 Chapter 1 Vector Spaces Linear algebra is the study of linear maps on finite-dimensional vector spaces. Eventually we will learn what all these terms mean. In this chapter we will define vector spaces

More information

ENTRY GROUP THEORY. [ENTRY GROUP THEORY] Authors: started Mark Lezama: October 2003 Literature: Algebra by Michael Artin, Mathworld.

ENTRY GROUP THEORY. [ENTRY GROUP THEORY] Authors: started Mark Lezama: October 2003 Literature: Algebra by Michael Artin, Mathworld. ENTRY GROUP THEORY [ENTRY GROUP THEORY] Authors: started Mark Lezama: October 2003 Literature: Algebra by Michael Artin, Mathworld Group theory [Group theory] is studies algebraic objects called groups.

More information

Definition List Modern Algebra, Fall 2011 Anders O.F. Hendrickson

Definition List Modern Algebra, Fall 2011 Anders O.F. Hendrickson Definition List Modern Algebra, Fall 2011 Anders O.F. Hendrickson On almost every Friday of the semester, we will have a brief quiz to make sure you have memorized the definitions encountered in our studies.

More information

Representations of quivers

Representations of quivers Representations of quivers Gwyn Bellamy October 13, 215 1 Quivers Let k be a field. Recall that a k-algebra is a k-vector space A with a bilinear map A A A making A into a unital, associative ring. Notice

More information

Computational Group Theory

Computational Group Theory Computational Group Theory Soria Summer School 2009 Session 3: Coset enumeration July 2009, Hans Sterk (sterk@win.tue.nl) Where innovation starts Coset enumeration: contents 2/25 What is coset enumeration

More information

Exercises on chapter 0

Exercises on chapter 0 Exercises on chapter 0 1. A partially ordered set (poset) is a set X together with a relation such that (a) x x for all x X; (b) x y and y x implies that x = y for all x, y X; (c) x y and y z implies that

More information

where Σ is a finite discrete Gal(K sep /K)-set unramified along U and F s is a finite Gal(k(s) sep /k(s))-subset

where Σ is a finite discrete Gal(K sep /K)-set unramified along U and F s is a finite Gal(k(s) sep /k(s))-subset Classification of quasi-finite étale separated schemes As we saw in lecture, Zariski s Main Theorem provides a very visual picture of quasi-finite étale separated schemes X over a henselian local ring

More information

Aperiodic languages and generalizations

Aperiodic languages and generalizations Aperiodic languages and generalizations Lila Kari and Gabriel Thierrin Department of Mathematics University of Western Ontario London, Ontario, N6A 5B7 Canada June 18, 2010 Abstract For every integer k

More information

Chapter 1 The Real Numbers

Chapter 1 The Real Numbers Chapter 1 The Real Numbers In a beginning course in calculus, the emphasis is on introducing the techniques of the subject;i.e., differentiation and integration and their applications. An advanced calculus

More information

Axioms of Kleene Algebra

Axioms of Kleene Algebra Introduction to Kleene Algebra Lecture 2 CS786 Spring 2004 January 28, 2004 Axioms of Kleene Algebra In this lecture we give the formal definition of a Kleene algebra and derive some basic consequences.

More information

1 Chapter 6 - Exercise 1.8.cf

1 Chapter 6 - Exercise 1.8.cf 1 CHAPTER 6 - EXERCISE 1.8.CF 1 1 Chapter 6 - Exercise 1.8.cf Determine 1 The Class Equation of the dihedral group D 5. Note first that D 5 = 10 = 5 2. Hence every conjugacy class will have order 1, 2

More information

Math 210B. Profinite group cohomology

Math 210B. Profinite group cohomology Math 210B. Profinite group cohomology 1. Motivation Let {Γ i } be an inverse system of finite groups with surjective transition maps, and define Γ = Γ i equipped with its inverse it topology (i.e., the

More information

CHAPTER 7. Connectedness

CHAPTER 7. Connectedness CHAPTER 7 Connectedness 7.1. Connected topological spaces Definition 7.1. A topological space (X, T X ) is said to be connected if there is no continuous surjection f : X {0, 1} where the two point set

More information

1 p mr. r, where p 1 < p 2 < < p r are primes, reduces then to the problem of finding, for i = 1,...,r, all possible partitions (p e 1

1 p mr. r, where p 1 < p 2 < < p r are primes, reduces then to the problem of finding, for i = 1,...,r, all possible partitions (p e 1 Theorem 2.9 (The Fundamental Theorem for finite abelian groups). Let G be a finite abelian group. G can be written as an internal direct sum of non-trival cyclic groups of prime power order. Furthermore

More information

TROPICAL SCHEME THEORY

TROPICAL SCHEME THEORY TROPICAL SCHEME THEORY 5. Commutative algebra over idempotent semirings II Quotients of semirings When we work with rings, a quotient object is specified by an ideal. When dealing with semirings (and lattices),

More information

Commutative Banach algebras 79

Commutative Banach algebras 79 8. Commutative Banach algebras In this chapter, we analyze commutative Banach algebras in greater detail. So we always assume that xy = yx for all x, y A here. Definition 8.1. Let A be a (commutative)

More information

Free Subgroups of the Fundamental Group of the Hawaiian Earring

Free Subgroups of the Fundamental Group of the Hawaiian Earring Journal of Algebra 219, 598 605 (1999) Article ID jabr.1999.7912, available online at http://www.idealibrary.com on Free Subgroups of the Fundamental Group of the Hawaiian Earring Katsuya Eda School of

More information

Supplement. Dr. Bob s Modern Algebra Glossary Based on Fraleigh s A First Course on Abstract Algebra, 7th Edition, Sections 0 through IV.

Supplement. Dr. Bob s Modern Algebra Glossary Based on Fraleigh s A First Course on Abstract Algebra, 7th Edition, Sections 0 through IV. Glossary 1 Supplement. Dr. Bob s Modern Algebra Glossary Based on Fraleigh s A First Course on Abstract Algebra, 7th Edition, Sections 0 through IV.23 Abelian Group. A group G, (or just G for short) is

More information