2.3 The Krull-Schmidt Theorem
|
|
- Lambert Curtis
- 5 years ago
- Views:
Transcription
1 2.3. THE KRULL-SCHMIDT THEOREM The Krull-Schmidt Theorem Every finitely generated abelian group is a direct sum of finitely many indecomposable abelian groups Z and Z p n. We will study a large class of (not necessarily abelian) groups which can be decomposed into a direct product of indecomposable groups. Def. A group G is indecomposable if G {e} and G is not the direct product of two of its proper subgroups. Every simple group is indecomposable (e.g. A n for n 4, Z p for prime p). However indecomposable groups need not be simple (e.g. Z, Z p n for n 2, and S n ) Def. Let G be a group. G satisfies the ascending chain condition (ACC) if for every chain G 1 < G 2 < of normal subgroups of G there is an integer n such that G i = G n for all i > n; G satisfies the descending chain condition (DCC) if for every chain G 1 > G 2 > of normal subgroups of G there is an integer n such that G i = G n for all i > n. Ex. Every finite group satisfies both chain conditions. Ex. (HW) Z satisfies ACC but not DCC. Ex. (HW) Z(p ) = {a/b Q/Z b = p n for some n N} satisfies DCC but not ACC. Thm If G satisfies either ACC or DCC, then G is a direct product of a finite number of indecomposable subgroups. Proof. Call a group good if it is a direct product of a finite number of indecomposable subgroups; call it bad otherwise. Facts: 1. Indecomposable groups are good. 2. If P = H K and both H and K are good, then P is good. Suppose on the contrary, G is bad. Then G is not indecomposable. So G = H 1 K 1 for some H 1, K 1 G. At least one of H 1 and K 1 should be bad. Without lost of generality, assume that K 1 is bad. Then K 1 = H 2 K 2 for H 2, K 2 K 1, where at least one of H 2 and K 2 is bad, say K 2 is bad. Repeating the process, we have G = H 1 (H 2 (H 3 ) ) H 1 H 2 H 3 There are an infinite strictly ascending chain and an infinite strictly descending chain of G: H 1 < H 1 H 2 < H 1 H 2 H 3 < and G > K 1 > K 2 > These violate both ACC and DCC. Therefore G is good.
2 42 CHAPTER 2. THE STRUCTURE OF GROUPS Def. An endomorphism f of a group G is called a normal endomorphism if af(b)a 1 = f(aba 1 ) for all a, b G. Lem If ϕ and ψ are normal endomorphisms of a group G, then so is ϕ ψ. 2. If ϕ is a normal endomorphism of G and if H G, then ϕ(h) G. 3. If ϕ is a normal automorphism of a group G, then ϕ 1 is also normal. Def. If ϕ and ψ are endomorphisms of G, then ϕ + ψ : G G is the function defined by x ϕ(x)ψ(x). For general group G, ϕ+ψ is not necessarily commutative on +, and it is not necessarily a homomorphism. Lem Let G = H 1 H m have projections π i : G H i and inclusion ι i : H i G. Then the sum of any k distinct ι i π i is a normal endomorphism of G. Moreover, the sum of all the ι i π i is the identity function on G. Proof. If ϕ = k i=1 ι iπ i, then ϕ(h 1,, h m ) = h 1 h k, that is, ϕ = ιπ, where π is the projection of G onto the direct factor H 1 H k and λ is the inclusion of H 1 H k into G, It follows that ϕ is a normal endomorphism of G and, if k = m, that ϕ = 1 G. Lem Let G have both chain conditions. If ϕ is a normal endomorphism of G, then ϕ is an injection if and only if it is a surjection. (Thus, either property ensures that ϕ is an automorphism.) Proof. Suppose that ϕ is an injection and that g ϕ(g). Then ϕ n is an injection, and thus ϕ n (g) ϕ n (ϕ(g)) = ϕ n+1 (G). There is a stricly descending chain of subgroups G > ϕ(g) > ϕ 2 (G) >. Now ϕ normal implies ϕ n is normal. By Lemma 2.14, ϕ n (G) G for all n, and so the DCC is violated. Therefore, ϕ is a surjection. Assume that ϕ is a surjection. Then ϕ n is a surjection for all n. We claim that Ker ϕ = {e}. Otherwise, there is x e and x Ker ϕ. Write x = ϕ n (g) for some g G. Then g Ker ϕ n but g Ker ϕ n+1. Thus we have a strictly ascending chain of normal subgroups {e} = Ker ϕ 0 < Ker ϕ 1 < Ker ϕ 2 <. So the ACC is violated. Thus ϕ is an injection.
3 2.3. THE KRULL-SCHMIDT THEOREM 43 Lem 2.17 (Fitting s Lemma). Let G have both chain conditions and ϕ a normal endomorphism of G. Then G = Ker ϕ n Im ϕ n for some n 1. Proof. Since ϕ is normal, we have the following two chains of normal subgorups G Im ϕ 1 Im ϕ 2 and {e} Ker ϕ 1 Ker ϕ 2 By hypothesis there is an n such that Im ϕ k = Im ϕ n and Ker ϕ k = Ker ϕ n for all k n. Claim: Ker ϕ n Im ϕ n = {e}. Otherwise, there is x e and x Ker ϕ n Im ϕ n. Let x = ϕ n (g). Then ϕ 2n (g) = ϕ n (x) = e. So g Ker ϕ 2n = Ker ϕ n. So x = ϕ n (g) = e, a contradiction. Thus Ker ϕ n Im ϕ n = {e}. Given x G, ϕ n (x) = ϕ 2n (g) by Im (ϕ n ) = Im (ϕ 2n ). So x = kϕ n (g) for some k Ker ϕ n. This shows that G = Ker ϕ n Im ϕ n. Therefore G = Ker ϕ n Im ϕ n. Def. An endomorphism ϕ of a group G is nilpotent if ϕ n (G) = {e} for some n. Cor If G is an indecomposable group having both chain conditions, then every normal endomorphism ϕ of G is either nilpotent or an automorphism. Proof. There is n 1 such that G = Ker ϕ n Im ϕ n. Then either Ker ϕ n = {e} or Im ϕ n = {e} since G is indecomposable. If Ker ϕ n = {e}, then Ker ϕ = {e}, and thus ϕ is an injection, and thus ϕ is an automorphism by Lemma If Im ϕ n = {e}, then ϕ is nilpotent. Lem Let G be an indecomposable group with both chain conditions, and let ϕ and ψ be normal nilpotent endomorphisms of G. If ϕ + ψ is an endomorphism of G, then it is normal nilpotent. Proof. a(ϕ + ψ)(b)a 1 = aϕ(b)a 1 + bψ(b)a 1 = ϕ(aba 1 ) + ψ(aba 1 ) = (ϕ + ψ)(aba 1 ). If ϕ + ψ is an endomorphism, then it is normal. By Lemma 2.18, ϕ + ψ is either nilpotent or an automorphism. If ϕ + ψ is an automorphism, then Lemma 2.14 says that its inverse γ is also normal. One has 1 G = (ϕ + ψ)γ = ϕγ + ψγ where ϕγ := ϕ γ. Let λ = ϕγ and µ = ψγ. Then 1 G = λ + µ. So x 1 = λ(x 1 )µ(x 1 ) and, taking inverses, x = µ(x)λ(x); that is, λ + µ = µ + λ. The equation λ(λ + µ) = (λ + µ)λ implies that λµ = µλ. Then for any integer m > 0, (λ + µ) n = ( ) m λ i µ m i. (2.2) i i Nilpotence of ϕ and ψ implies nilpotence of λ = ϕγ and µ = ψγ (they cannot be automorphisms because they have nontrivial kernels); there are r, s Z + with λ r = 0 and µ 2 = 0. If m = r + s 1, then (2.2) shows that 1 G = (λ + µ) m = 0, forcing G = {e}. This is a contradiction, for every indecomposable group is nontrivial.
4 44 CHAPTER 2. THE STRUCTURE OF GROUPS Cor Let G be an indecomposable group having both chain conditions. If ϕ 1,, ϕ n is a set of normal nilpotent endomorphisms of G such that every sum of distinct ϕ s is an endomorphism, then ϕ ϕ n is nilpotent. Proof. Induction on n. Thm 2.21 (Krull-Schmidt). Let G be a group having both ACC and DCC. If G = H 1 H s = K 1 K t are two decompositions of G into indecomposable factors, then s = t and there is a reindexing of K s so that H i K i for all i. Moreover, given any r between 1 and s, the reindexing may be chosen so that G = H 1 H r K r+1 K s. Remark. The last conclusion is stronger than saying that the factors are determined up to isomorphism; one can replace factors of one decomposition by suitable factors from the other. Proof. We shall give the proof when r = 1; the reader may complete the proof for general r by inducton. Given the first decomposition, we must find a reindexing of the K s so that H i K i for all i and G = H 1 K 2 K t. Let π i : G H i and ι i : H i G be the projection and inclusion from G = H 1 H s, and π i : G K i and ι i : K i G the projection and inclusion from G = K 1 K t. The maps ι i π i and ι i π i are normal endomorphisms of G. By Lemma 2.15, every partial sum ι i π i is a normal endomorphism of G. Hence, every partial sum of 1 H1 = π 1 ι 1 = π 1 1 G ι 1 = π 1 ( ι jπ j)ι 1 = π 1 ι jπ jι 1 is a normal endomorphism of H 1. Since 1 H1 is not nilpotent and H 1 satisfies both chain conditions (as G does and every normal subgroup of H 1 is also normal in G), Corollary 2.20 gives an index j with π 1 ι j π j ι 1 an automorphism. We reindex so that π 1 ι 1 π 1 ι 1 is an automorphism of H 1 ; let γ be its inverse. We claim that π 1 ι 1 : H 1 K 1 is an isomorphism. We have (γπ 1 ι 1 )(π 1 ι 1) = 1 H1. Let us show that θ = (π 1 ι 1)(γπ 1 ι 1 ) = 1 K 1. First, θ 2 = θ. Second, 1 H1 = 1 H1 1 H1 = (γπ 1 ι 1)(π 1ι 1 )(γπ 1 ι 1)(π 1ι 1 ) = (γπ 1 ι 1)θ(π 1ι 1 ), so that θ 0. Were θ nilpotent, then θ 2 = θ forces θ = 0. Therefore, θ is an automorphism of K 1, and so θ 2 = θ gives θ = 1 K1. Thus π 1 ι 1 : H 1 K 1 is an isomorphism. Now π 1 sends K 2 K t into {e} while π 1 ι 1 restricts to an isomorphism on H 1. Therefore H 1 (K 2 K t ) = {e}. Then G := H 1, K 2 K t = H 1 K 2 K t.
5 2.3. THE KRULL-SCHMIDT THEOREM 45 If x G, then x = k 1 k 2 k t, where k j K j. Since π 1 ι 1 is an isomorphism, the map β : G G, defined by x π 1 ι 1 (k 1)k 2 k t, is an injection with image G. By Lemma 2.16, β is a surjection; that is, G = G = H 1 K 2 K t. Finally, K 2 K t G/H 1 H 2 H s, so that the remaining uniqueness assertions follow by induction on max{s, t}.
II. Products of Groups
II. Products of Groups Hong-Jian Lai October 2002 1. Direct Products (1.1) The direct product (also refereed as complete direct sum) of a collection of groups G i, i I consists of the Cartesian product
More informationLecture 2. (1) Every P L A (M) has a maximal element, (2) Every ascending chain of submodules stabilizes (ACC).
Lecture 2 1. Noetherian and Artinian rings and modules Let A be a commutative ring with identity, A M a module, and φ : M N an A-linear map. Then ker φ = {m M : φ(m) = 0} is a submodule of M and im φ is
More information4.3 Composition Series
4.3 Composition Series Let M be an A-module. A series for M is a strictly decreasing sequence of submodules M = M 0 M 1... M n = {0} beginning with M and finishing with {0 }. The length of this series
More information4.2 Chain Conditions
4.2 Chain Conditions Imposing chain conditions on the or on the poset of submodules of a module, poset of ideals of a ring, makes a module or ring more tractable and facilitates the proofs of deep theorems.
More informationMath 594. Solutions 5
Math 594. Solutions 5 Book problems 6.1: 7. Prove that subgroups and quotient groups of nilpotent groups are nilpotent (your proof should work for infinite groups). Give an example of a group G which possesses
More informationAlgebra. Travis Dirle. December 4, 2016
Abstract Algebra 2 Algebra Travis Dirle December 4, 2016 2 Contents 1 Groups 1 1.1 Semigroups, Monoids and Groups................ 1 1.2 Homomorphisms and Subgroups................. 2 1.3 Cyclic Groups...........................
More informationFrank Moore Algebra 901 Notes Professor: Tom Marley Direct Products of Groups:
Frank Moore Algebra 901 Notes Professor: Tom Marley Direct Products of Groups: Definition: The external direct product is defined to be the following: Let H 1,..., H n be groups. H 1 H 2 H n := {(h 1,...,
More information12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold.
12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold. 12.1. Indecomposability of M and the localness of End
More informationINJECTIVE MODULES: PREPARATORY MATERIAL FOR THE SNOWBIRD SUMMER SCHOOL ON COMMUTATIVE ALGEBRA
INJECTIVE MODULES: PREPARATORY MATERIAL FOR THE SNOWBIRD SUMMER SCHOOL ON COMMUTATIVE ALGEBRA These notes are intended to give the reader an idea what injective modules are, where they show up, and, to
More informationREPRESENTATION THEORY WEEK 9
REPRESENTATION THEORY WEEK 9 1. Jordan-Hölder theorem and indecomposable modules Let M be a module satisfying ascending and descending chain conditions (ACC and DCC). In other words every increasing sequence
More informationHonors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35
Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35 1. Let R be a commutative ring with 1 0. (a) Prove that the nilradical of R is equal to the intersection of the prime
More informationA Krull-Schmidt Theorem for Noetherian Modules *
A Krull-Schmidt Theorem for Noetherian Modules * Gary Brookfield Department of Mathematics, University of California, Riverside CA 92521-0135 E-mail: brookfield@math.ucr.edu We prove a version of the Krull-Schmidt
More informationElements 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 informationA 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 informationRohit 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 informationA 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 informationA COURSE IN HOMOLOGICAL ALGEBRA CHAPTER 11: Auslander s Proof of Roiter s Theorem E. L. Lady (April 29, 1998)
A COURSE IN HOMOLOGICAL ALGEBRA CHAPTER 11: Auslander s Proof of Roiter s Theorem E. L. Lady (April 29, 1998) A category C is skeletally small if there exists a set of objects in C such that every object
More informationMath 120 HW 9 Solutions
Math 120 HW 9 Solutions June 8, 2018 Question 1 Write down a ring homomorphism (no proof required) f from R = Z[ 11] = {a + b 11 a, b Z} to S = Z/35Z. The main difficulty is to find an element x Z/35Z
More informationStructure of rings. Chapter Algebras
Chapter 5 Structure of rings 5.1 Algebras It is time to introduce the notion of an algebra over a commutative ring. So let R be a commutative ring. An R-algebra is a ring A (unital as always) together
More informationA SIMPLE PROOF OF BURNSIDE S CRITERION FOR ALL GROUPS OF ORDER n TO BE CYCLIC
A SIMPLE PROOF OF BURNSIDE S CRITERION FOR ALL GROUPS OF ORDER n TO BE CYCLIC SIDDHI PATHAK Abstract. This note gives a simple proof of a famous theorem of Burnside, namely, all groups of order n are cyclic
More informationCHAPTER 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 informationKevin James. p-groups, Nilpotent groups and Solvable groups
p-groups, Nilpotent groups and Solvable groups Definition A maximal subgroup of a group G is a proper subgroup M G such that there are no subgroups H with M < H < G. Definition A maximal subgroup of a
More informationφ(xy) = (xy) n = x n y n = φ(x)φ(y)
Groups 1. (Algebra Comp S03) Let A, B and C be normal subgroups of a group G with A B. If A C = B C and AC = BC then prove that A = B. Let b B. Since b = b1 BC = AC, there are a A and c C such that b =
More informationALGEBRAIC 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 informationMath 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 informationAbstract Algebra II Groups ( )
Abstract Algebra II Groups ( ) Melchior Grützmann / melchiorgfreehostingcom/algebra October 15, 2012 Outline Group homomorphisms Free groups, free products, and presentations Free products ( ) Definition
More informationGroups 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 informationA PROOF OF BURNSIDE S p a q b THEOREM
A PROOF OF BURNSIDE S p a q b THEOREM OBOB Abstract. We prove that if p and q are prime, then any group of order p a q b is solvable. Throughout this note, denote by A the set of algebraic numbers. We
More information(5.11) (Second Isomorphism Theorem) If K G and N G, then K/(N K) = NK/N. PF: Verify N HK. Find a homomorphism f : K HK/N with ker(f) = (N K).
Lecture Note of Week 3 6. Normality, Quotients and Homomorphisms (5.7) A subgroup N satisfying any one properties of (5.6) is called a normal subgroup of G. Denote this fact by N G. The homomorphism π
More informationMA441: Algebraic Structures I. Lecture 26
MA441: Algebraic Structures I Lecture 26 10 December 2003 1 (page 179) Example 13: A 4 has no subgroup of order 6. BWOC, suppose H < A 4 has order 6. Then H A 4, since it has index 2. Thus A 4 /H has order
More informationTHE LENGTH OF NOETHERIAN MODULES
THE LENGTH OF NOETHERIAN MODULES GARY BROOKFIELD Abstract. We define an ordinal valued length for Noetherian modules which extends the usual definition of composition series length for finite length modules.
More informationCourse 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 informationWinter 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 informationALGEBRA EXERCISES, PhD EXAMINATION LEVEL
ALGEBRA EXERCISES, PhD EXAMINATION LEVEL 1. Suppose that G is a finite group. (a) Prove that if G is nilpotent, and H is any proper subgroup, then H is a proper subgroup of its normalizer. (b) Use (a)
More informationTwo subgroups and semi-direct products
Two subgroups and semi-direct products 1 First remarks Throughout, we shall keep the following notation: G is a group, written multiplicatively, and H and K are two subgroups of G. We define the subset
More informationInjective Modules and Matlis Duality
Appendix A Injective Modules and Matlis Duality Notes on 24 Hours of Local Cohomology William D. Taylor We take R to be a commutative ring, and will discuss the theory of injective R-modules. The following
More informationThe Ascending Chain Condition. E. L. Lady. Every surjective endomorphism of M is an automorphism.
The Ascending Chain Condition E. L. Lady The goal is to find a condition on an R-module M which will ensure the following property: ( ) Every surjective endomorphism of M is an automorphism. To start with,
More informationREPRESENTATION THEORY, LECTURE 0. BASICS
REPRESENTATION THEORY, LECTURE 0. BASICS IVAN LOSEV Introduction The aim of this lecture is to recall some standard basic things about the representation theory of finite dimensional algebras and finite
More informationSolutions 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 informationMath 121 Homework 4: Notes on Selected Problems
Math 121 Homework 4: Notes on Selected Problems 11.2.9. If W is a subspace of the vector space V stable under the linear transformation (i.e., (W ) W ), show that induces linear transformations W on W
More informationMath 581 Problem Set 8 Solutions
Math 581 Problem Set 8 Solutions 1. Prove that a group G is abelian if and only if the function ϕ : G G given by ϕ(g) g 1 is a homomorphism of groups. In this case show that ϕ is an isomorphism. Proof:
More informationD-MATH Algebra II FS18 Prof. Marc Burger. Solution 21. Solvability by Radicals
D-MATH Algebra II FS18 Prof. Marc Burger Solution 21 Solvability by Radicals 1. Let be (N, ) and (H, ) be two groups and ϕ : H Aut(N) a group homomorphism. Write ϕ h := ϕ(h) Aut(N) for each h H. Define
More information4.4 Noetherian Rings
4.4 Noetherian Rings Recall that a ring A is Noetherian if it satisfies the following three equivalent conditions: (1) Every nonempty set of ideals of A has a maximal element (the maximal condition); (2)
More informationALGEBRA QUALIFYING EXAM PROBLEMS
ALGEBRA QUALIFYING EXAM PROBLEMS Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND MODULES General
More informationRELATIVE EXT GROUPS, RESOLUTIONS, AND SCHANUEL CLASSES
RELATIVE EXT GROUPS, RESOLUTIONS, AND SCHANUEL CLASSES HENRIK HOLM Abstract. Given a precovering (also called contravariantly finite) class there are three natural approaches to a homological dimension
More informationMath 210C. The representation ring
Math 210C. The representation ring 1. Introduction Let G be a nontrivial connected compact Lie group that is semisimple and simply connected (e.g., SU(n) for n 2, Sp(n) for n 1, or Spin(n) for n 3). Let
More informationTopics in Module Theory
Chapter 7 Topics in Module Theory This chapter will be concerned with collecting a number of results and constructions concerning modules over (primarily) noncommutative rings that will be needed to study
More informationHOMOLOGY 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 information1.5 The Nil and Jacobson Radicals
1.5 The Nil and Jacobson Radicals The idea of a radical of a ring A is an ideal I comprising some nasty piece of A such that A/I is well-behaved or tractable. Two types considered here are the nil and
More informationBASIC 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 informationProjective modules: Wedderburn rings
Projective modules: Wedderburn rings April 10, 2008 8 Wedderburn rings A Wedderburn ring is an artinian ring which has no nonzero nilpotent left ideals. Note that if R has no left ideals I such that I
More informationNOTES ON FINITE FIELDS
NOTES ON FINITE FIELDS AARON LANDESMAN CONTENTS 1. Introduction to finite fields 2 2. Definition and constructions of fields 3 2.1. The definition of a field 3 2.2. Constructing field extensions by adjoining
More informationON THE RESIDUALITY A FINITE p-group OF HN N-EXTENSIONS
1 ON THE RESIDUALITY A FINITE p-group OF HN N-EXTENSIONS D. I. Moldavanskii arxiv:math/0701498v1 [math.gr] 18 Jan 2007 A criterion for the HNN-extension of a finite p-group to be residually a finite p-group
More informationGROUPS OF ORDER p 3 KEITH CONRAD
GROUPS OF ORDER p 3 KEITH CONRAD For any prime p, we want to describe the groups of order p 3 up to isomorphism. From the cyclic decomposition of finite abelian groups, there are three abelian groups of
More informationMath 210C. A non-closed commutator subgroup
Math 210C. A non-closed commutator subgroup 1. Introduction In Exercise 3(i) of HW7 we saw that every element of SU(2) is a commutator (i.e., has the form xyx 1 y 1 for x, y SU(2)), so the same holds for
More informationKrull Dimension and Going-Down in Fixed Rings
David Dobbs Jay Shapiro April 19, 2006 Basics R will always be a commutative ring and G a group of (ring) automorphisms of R. We let R G denote the fixed ring, that is, Thus R G is a subring of R R G =
More informationTEST ELEMENTS IN TORSION-FREE HYPERBOLIC GROUPS
TEST ELEMENTS IN TORSION-FREE HYPERBOLIC GROUPS DANIEL GROVES Abstract. We prove that in a torsion-free hyperbolic group, an element is a test element if and only if it is not contained in a proper retract.
More informationMATH 326: RINGS AND MODULES STEFAN GILLE
MATH 326: RINGS AND MODULES STEFAN GILLE 1 2 STEFAN GILLE 1. Rings We recall first the definition of a group. 1.1. Definition. Let G be a non empty set. The set G is called a group if there is a map called
More informationPRIMARY DECOMPOSITION OF MODULES
PRIMARY DECOMPOSITION OF MODULES SUDHIR R. GHORPADE AND JUGAL K. VERMA Department of Mathematics, Indian Institute of Technology, Mumbai 400 076, India E-Mail: {srg, jkv}@math.iitb.ac.in Abstract. Basics
More informationMATRIX LIE GROUPS AND LIE GROUPS
MATRIX LIE GROUPS AND LIE GROUPS Steven Sy December 7, 2005 I MATRIX LIE GROUPS Definition: A matrix Lie group is a closed subgroup of Thus if is any sequence of matrices in, and for some, then either
More informationChapter 2 Linear Transformations
Chapter 2 Linear Transformations Linear Transformations Loosely speaking, a linear transformation is a function from one vector space to another that preserves the vector space operations. Let us be more
More informationWIDE SUBCATEGORIES OF d-cluster TILTING SUBCATEGORIES
WIDE SUBCATEGORIES OF d-cluster TILTING SUBCATEGORIES MARTIN HERSCHEND, PETER JØRGENSEN, AND LAERTIS VASO Abstract. A subcategory of an abelian category is wide if it is closed under sums, summands, kernels,
More informationSection III.15. Factor-Group Computations and Simple Groups
III.15 Factor-Group Computations 1 Section III.15. Factor-Group Computations and Simple Groups Note. In this section, we try to extract information about a group G by considering properties of the factor
More informationDefinitions. 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 informationMINIMAL NUMBER OF GENERATORS AND MINIMUM ORDER OF A NON-ABELIAN GROUP WHOSE ELEMENTS COMMUTE WITH THEIR ENDOMORPHIC IMAGES
Communications in Algebra, 36: 1976 1987, 2008 Copyright Taylor & Francis roup, LLC ISSN: 0092-7872 print/1532-4125 online DOI: 10.1080/00927870801941903 MINIMAL NUMBER OF ENERATORS AND MINIMUM ORDER OF
More informationExercises MAT2200 spring 2013 Ark 4 Homomorphisms and factor groups
Exercises MAT2200 spring 2013 Ark 4 Homomorphisms and factor groups This Ark concerns the weeks No. (Mar ) and No. (Mar ). Plans until Eastern vacations: In the book the group theory included in the curriculum
More informationCONSEQUENCES 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 informationAlgebra-I, Fall Solutions to Midterm #1
Algebra-I, Fall 2018. Solutions to Midterm #1 1. Let G be a group, H, K subgroups of G and a, b G. (a) (6 pts) Suppose that ah = bk. Prove that H = K. Solution: (a) Multiplying both sides by b 1 on the
More informationMath 210B: Algebra, Homework 4
Math 210B: Algebra, Homework 4 Ian Coley February 5, 2014 Problem 1. Let S be a multiplicative subset in a commutative ring R. Show that the localisation functor R-Mod S 1 R-Mod, M S 1 M, is exact. First,
More informationAlgebra SEP Solutions
Algebra SEP Solutions 17 July 2017 1. (January 2017 problem 1) For example: (a) G = Z/4Z, N = Z/2Z. More generally, G = Z/p n Z, N = Z/pZ, p any prime number, n 2. Also G = Z, N = nz for any n 2, since
More informationA GRAPHICAL REPRESENTATION OF RINGS VIA AUTOMORPHISM GROUPS
A GRAPHICAL REPRESENTATION OF RINGS VIA AUTOMORPHISM GROUPS N. MOHAN KUMAR AND PRAMOD K. SHARMA Abstract. Let R be a commutative ring with identity. We define a graph Γ Aut R (R) on R, with vertices elements
More informationMath 429/581 (Advanced) Group Theory. Summary of Definitions, Examples, and Theorems by Stefan Gille
Math 429/581 (Advanced) Group Theory Summary of Definitions, Examples, and Theorems by Stefan Gille 1 2 0. Group Operations 0.1. Definition. Let G be a group and X a set. A (left) operation of G on X is
More informationSection II.1. Free Abelian Groups
II.1. Free Abelian Groups 1 Section II.1. Free Abelian Groups Note. This section and the next, are independent of the rest of this chapter. The primary use of the results of this chapter is in the proof
More informationMATH 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 informationTree-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 informationAlgebraic Structures II
MAS 305 Algebraic Structures II Notes 9 Autumn 2006 Composition series soluble groups Definition A normal subgroup N of a group G is called a maximal normal subgroup of G if (a) N G; (b) whenever N M G
More information(1) A frac = b : a, b A, b 0. We can define addition and multiplication of fractions as we normally would. a b + c d
The Algebraic Method 0.1. Integral Domains. Emmy Noether and others quickly realized that the classical algebraic number theory of Dedekind could be abstracted completely. In particular, rings of integers
More informationNOTES FOR COMMUTATIVE ALGEBRA M5P55
NOTES FOR COMMUTATIVE ALGEBRA M5P55 AMBRUS PÁL 1. Rings and ideals Definition 1.1. A quintuple (A, +,, 0, 1) is a commutative ring with identity, if A is a set, equipped with two binary operations; addition
More informationINVERSE 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 informationIntroduction to Groups
Introduction to Groups Hong-Jian Lai August 2000 1. Basic Concepts and Facts (1.1) A semigroup is an ordered pair (G, ) where G is a nonempty set and is a binary operation on G satisfying: (G1) a (b c)
More informationits image and kernel. A subgroup of a group G is a non-empty subset K of G such that k 1 k 1
10 Chapter 1 Groups 1.1 Isomorphism theorems Throughout the chapter, we ll be studying the category of groups. Let G, H be groups. Recall that a homomorphism f : G H means a function such that f(g 1 g
More informationON VARIETIES IN WHICH SOLUBLE GROUPS ARE TORSION-BY-NILPOTENT
ON VARIETIES IN WHICH SOLUBLE GROUPS ARE TORSION-BY-NILPOTENT GÉRARD ENDIMIONI C.M.I., Université de Provence, UMR-CNRS 6632 39, rue F. Joliot-Curie, 13453 Marseille Cedex 13, France E-mail: endimion@gyptis.univ-mrs.fr
More informationRational Homotopy Theory Seminar Week 11: Obstruction theory for rational homotopy equivalences J.D. Quigley
Rational Homotopy Theory Seminar Week 11: Obstruction theory for rational homotopy equivalences J.D. Quigley Reference. Halperin-Stasheff Obstructions to homotopy equivalences Question. When can a given
More informationMath 103 HW 9 Solutions to Selected Problems
Math 103 HW 9 Solutions to Selected Problems 4. Show that U(8) is not isomorphic to U(10). Solution: Unfortunately, the two groups have the same order: the elements are U(n) are just the coprime elements
More informationThe Cartan Decomposition of a Complex Semisimple Lie Algebra
The Cartan Decomposition of a Complex Semisimple Lie Algebra Shawn Baland University of Colorado, Boulder November 29, 2007 Definition Let k be a field. A k-algebra is a k-vector space A equipped with
More informationInfinite-Dimensional Triangularization
Infinite-Dimensional Triangularization Zachary Mesyan March 11, 2018 Abstract The goal of this paper is to generalize the theory of triangularizing matrices to linear transformations of an arbitrary vector
More informationNOTES 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 informationMATH 436 Notes: Cyclic groups and Invariant Subgroups.
MATH 436 Notes: Cyclic groups and Invariant Subgroups. Jonathan Pakianathan September 30, 2003 1 Cyclic Groups Now that we have enough basic tools, let us go back and study the structure of cyclic groups.
More informationGroups 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 informationFormal power series rings, inverse limits, and I-adic completions of rings
Formal power series rings, inverse limits, and I-adic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely
More informationNon-separable AF-algebras
Non-separable AF-algebras Takeshi Katsura Department of Mathematics, Hokkaido University, Kita 1, Nishi 8, Kita-Ku, Sapporo, 6-81, JAPAN katsura@math.sci.hokudai.ac.jp Summary. We give two pathological
More informationSmith theory. Andrew Putman. Abstract
Smith theory Andrew Putman Abstract We discuss theorems of P. Smith and Floyd connecting the cohomology of a simplicial complex equipped with an action of a finite p-group to the cohomology of its fixed
More informationMath 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 informationMATH 221 NOTES BRENT HO. Date: January 3, 2009.
MATH 22 NOTES BRENT HO Date: January 3, 2009. 0 Table of Contents. Localizations......................................................................... 2 2. Zariski Topology......................................................................
More informationPROBLEMS FROM GROUP THEORY
PROBLEMS FROM GROUP THEORY Page 1 of 12 In the problems below, G, H, K, and N generally denote groups. We use p to stand for a positive prime integer. Aut( G ) denotes the group of automorphisms of G.
More informationSolutions of exercise sheet 4
D-MATH Algebra I HS 14 Prof. Emmanuel Kowalski Solutions of exercise sheet 4 The content of the marked exercises (*) should be known for the exam. 1. Prove the following two properties of groups: 1. Every
More informationON STRONGLY PRIME IDEALS AND STRONGLY ZERO-DIMENSIONAL RINGS. Christian Gottlieb
ON STRONGLY PRIME IDEALS AND STRONGLY ZERO-DIMENSIONAL RINGS Christian Gottlieb Department of Mathematics, University of Stockholm SE-106 91 Stockholm, Sweden gottlieb@math.su.se Abstract A prime ideal
More informationGeneralized Alexander duality and applications. Osaka Journal of Mathematics. 38(2) P.469-P.485
Title Generalized Alexander duality and applications Author(s) Romer, Tim Citation Osaka Journal of Mathematics. 38(2) P.469-P.485 Issue Date 2001-06 Text Version publisher URL https://doi.org/10.18910/4757
More informationAlgebra Exam Fall Alexander J. Wertheim Last Updated: October 26, Groups Problem Problem Problem 3...
Algebra Exam Fall 2006 Alexander J. Wertheim Last Updated: October 26, 2017 Contents 1 Groups 2 1.1 Problem 1..................................... 2 1.2 Problem 2..................................... 2
More informationALGEBRA I (LECTURE NOTES 2017/2018) LECTURE 9 - CYCLIC GROUPS AND EULER S FUNCTION
ALGEBRA I (LECTURE NOTES 2017/2018) LECTURE 9 - CYCLIC GROUPS AND EULER S FUNCTION PAVEL RŮŽIČKA 9.1. Congruence modulo n. Let us have a closer look at a particular example of a congruence relation on
More information