BLECHER • All Numbers Below Are Integers, Usually Positive. • Read Up
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Classification of Finite Abelian Groups
Math 317 C1 John Sullivan Spring 2003 Classification of Finite Abelian Groups (Notes based on an article by Navarro in the Amer. Math. Monthly, February 2003.) The fundamental theorem of finite abelian groups expresses any such group as a product of cyclic groups: Theorem. Suppose G is a finite abelian group. Then G is (in a unique way) a direct product of cyclic groups of order pk with p prime. Our first step will be a special case of Cauchy’s Theorem, which we will prove later for arbitrary groups: whenever p |G| then G has an element of order p. Theorem (Cauchy). If G is a finite group, and p |G| is a prime, then G has an element of order p (or, equivalently, a subgroup of order p). ∼ Proof when G is abelian. First note that if |G| is prime, then G = Zp and we are done. In general, we work by induction. If G has no nontrivial proper subgroups, it must be a prime cyclic group, the case we’ve already handled. So we can suppose there is a nontrivial subgroup H smaller than G. Either p |H| or p |G/H|. In the first case, by induction, H has an element of order p which is also order p in G so we’re done. In the second case, if ∼ g + H has order p in G/H then |g + H| |g|, so hgi = Zkp for some k, and then kg ∈ G has order p. Note that we write our abelian groups additively. Definition. Given a prime p, a p-group is a group in which every element has order pk for some k. -
Cohomology Theory of Lie Groups and Lie Algebras
COHOMOLOGY THEORY OF LIE GROUPS AND LIE ALGEBRAS BY CLAUDE CHEVALLEY AND SAMUEL EILENBERG Introduction The present paper lays no claim to deep originality. Its main purpose is to give a systematic treatment of the methods by which topological questions concerning compact Lie groups may be reduced to algebraic questions con- cerning Lie algebras^). This reduction proceeds in three steps: (1) replacing questions on homology groups by questions on differential forms. This is accomplished by de Rham's theorems(2) (which, incidentally, seem to have been conjectured by Cartan for this very purpose); (2) replacing the con- sideration of arbitrary differential forms by that of invariant differential forms: this is accomplished by using invariant integration on the group manifold; (3) replacing the consideration of invariant differential forms by that of alternating multilinear forms on the Lie algebra of the group. We study here the question not only of the topological nature of the whole group, but also of the manifolds on which the group operates. Chapter I is concerned essentially with step 2 of the list above (step 1 depending here, as in the case of the whole group, on de Rham's theorems). Besides consider- ing invariant forms, we also introduce "equivariant" forms, defined in terms of a suitable linear representation of the group; Theorem 2.2 states that, when this representation does not contain the trivial representation, equi- variant forms are of no use for topology; however, it states this negative result in the form of a positive property of equivariant forms which is of interest by itself, since it is the key to Levi's theorem (cf. -
Lecture 1.3: Direct Products and Sums
Lecture 1.3: Direct products and sums Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 8530, Advanced Linear Algebra M. Macauley (Clemson) Lecture 1.3: Direct products and sums Math 8530, Advanced Linear Algebra 1 / 5 Overview In previous lectures, we learned about vectors spaces and subspaces. We learned about what it meant for a subset to span, to be linearly independent, and to be a basis. In this lecture, we will see how to create new vector spaces from old ones. We will see several ways to \multiply" vector spaces together, and will learn how to construct: the complement of a subspace the direct sum of two subspaces the direct product of two vector spaces M. Macauley (Clemson) Lecture 1.3: Direct products and sums Math 8530, Advanced Linear Algebra 2 / 5 Complements and direct sums Theorem 1.5 (a) Every subspace Y of a finite-dimensional vector space X is finite-dimensional. (b) Every subspace Y has a complement in X : another subspace Z such that every vector x 2 X can be written uniquely as x = y + z; y 2 Y ; z 2 Z; dim X = dim Y + dim Z: Proof Definition X is the direct sum of subspaces Y and Z that are complements of each other. More generally, X is the direct sum of subspaces Y1;:::; Ym if every x 2 X can be expressed uniquely as x = y1 + ··· + ym; yi 2 Yi : We denote this as X = Y1 ⊕ · · · ⊕ Ym. M. Macauley (Clemson) Lecture 1.3: Direct products and sums Math 8530, Advanced Linear Algebra 3 / 5 Direct products Definition The direct product of X1 and X2 is the vector space X1 × X2 := (x1; x2) j x1 2 X1; x2 2 X2 ; with addition and multiplication defined component-wise. -
Countable Torsion-Free Abelian Groups
COUNTABLE TORSION-FREE ABELIAN GROUPS By M. O'N. CAMPBELL [Received 27 January 1959.—Read 19 February 1959] Introduction IN this paper we present a new classification of the countable torsion-free abelian groups. Since any abelian group 0 may be regarded as an extension of a periodic group (namely the torsion part of G) by a torsion-free group, and since there exists the well-known complete classification of the count- able periodic abelian groups due to Ulm, it is clear that the classification problem studied here is of great interest. By a group we shall always mean an abelian group, and the additive notation will be employed throughout. It is well known that all the maximal linearly independent sets of elements of a given group are car- dinally equivalent; the corresponding cardinal number is the rank of the group. Derry (2) and Mal'cev (4) have studied the torsion-free groups of finite rank only, and have given classifications of these groups in terms of certain equivalence classes of systems of matrices with £>-adic elements. Szekeres (5) attempted to abolish the finiteness restriction on rank; he extracted certain invariants, but did not derive a complete classification. By a basis of a torsion-free group 0 we mean any maximal linearly independent subset of G. A subgroup generated by a basis of G will be called a basal subgroup or described as basal in G. Thus U is a basal sub- group of G if and only if (i) U is free abelian, and (ii) the factor group G/U is periodic. -
Arxiv:1909.05807V1 [Math.RA] 12 Sep 2019 Cs T
MODULES OVER TRUSSES VS MODULES OVER RINGS: DIRECT SUMS AND FREE MODULES TOMASZ BRZEZINSKI´ AND BERNARD RYBOLOWICZ Abstract. Categorical constructions on heaps and modules over trusses are con- sidered and contrasted with the corresponding constructions on groups and rings. These include explicit description of free heaps and free Abelian heaps, coprod- ucts or direct sums of Abelian heaps and modules over trusses, and description and analysis of free modules over trusses. It is shown that the direct sum of two non-empty Abelian heaps is always infinite and isomorphic to the heap associated to the direct sums of the group retracts of both heaps and Z. Direct sum is used to extend a given truss to a ring-type truss or a unital truss (or both). Free mod- ules are constructed as direct sums of a truss. It is shown that only free rank-one modules are free as modules over the associated truss. On the other hand, if a (finitely generated) module over a truss associated to a ring is free, then so is the corresponding quotient-by-absorbers module over this ring. 1. Introduction Trusses and skew trusses were defined in [3] in order to capture the nature of the distinctive distributive law that characterises braces and skew braces [12], [6], [9]. A (one-sided) truss is a set with a ternary operation which makes it into a heap or herd (see [10], [11], [1] or [13]) together with an associative binary operation that distributes (on one side or both) over the heap ternary operation. If the specific bi- nary operation admits it, a choice of a particular element could fix a group structure on a heap in a way that turns the truss into a ring or a brace-like system (which becomes a brace provided the binary operation admits inverses). -
Arxiv:2007.07508V1 [Math.GR] 15 Jul 2020 Re H Aoia Eopstosaeapplied
CANONICAL DECOMPOSITIONS OF ABELIAN GROUPS PHILL SCHULTZ Abstract. Every torsion–free abelian group of finite rank has two essentially unique complete direct decompositions whose summands come from specific classes of groups. 1. Introduction An intriguing feature of abelian group theory is that torsion–free groups, even those of finite rank, may fail to have unique complete decompositions; see for ex- ample [Fuchs, 2015, Chapter 12, §5] or [Mader and Schultz, 2018]). Therefore it is interesting to show that every such group has essentially unique complete de- compositions with summands from certain identifiable classes. For example, it was shown in [Mader and Schultz, 2018] that every finite rank torsion–free group G has a Main Decomposition, G = Gcd ⊕ Gcl where Gcd is completely decomposable and Gcl has no rank 1 direct summand; Gcd is unique up to isomorphism and Gcl unique up to near isomorphism. Let G be the class of torsion–free abelian groups of finite rank and let G ∈ G. The aim of this paper is to show that: • G = Gsd ⊕Gni where Gsd is a direct sum of strongly indecomposable groups and Gni has no strongly indecomposable direct summand. Gsd is unique up to isomorphism and Gni up to near isomorphism. • G = G(fq) ⊕ G(rq) ⊕ G(dq), where G(fq) is an extension of a completely decomposable group by a finite group, G(rq) is an extension of a completely decomposable group by an infinite reduced torsion group, and G(dq) is an extension of a completely decomposable group by an infinite torsion group which has a divisible summand. -
Chapter 3 Direct Sums, Affine Maps, the Dual Space, Duality
Chapter 3 Direct Sums, Affine Maps, The Dual Space, Duality 3.1 Direct Products, Sums, and Direct Sums There are some useful ways of forming new vector spaces from older ones. Definition 3.1. Given p 2vectorspacesE ,...,E , ≥ 1 p the product F = E1 Ep can be made into a vector space by defining addition⇥···⇥ and scalar multiplication as follows: (u1,...,up)+(v1,...,vp)=(u1 + v1,...,up + vp) λ(u1,...,up)=(λu1,...,λup), for all ui,vi Ei and all λ R. 2 2 With the above addition and multiplication, the vector space F = E E is called the direct product of 1 ⇥···⇥ p the vector spaces E1,...,Ep. 151 152 CHAPTER 3. DIRECT SUMS, AFFINE MAPS, THE DUAL SPACE, DUALITY The projection maps pr : E E E given by i 1 ⇥···⇥ p ! i pri(u1,...,up)=ui are clearly linear. Similarly, the maps in : E E E given by i i ! 1 ⇥···⇥ p ini(ui)=(0,...,0,ui, 0,...,0) are injective and linear. It can be shown (using bases) that dim(E E )=dim(E )+ +dim(E ). 1 ⇥···⇥ p 1 ··· p Let us now consider a vector space E and p subspaces U1,...,Up of E. We have a map a: U U E 1 ⇥···⇥ p ! given by a(u ,...,u )=u + + u , 1 p 1 ··· p with u U for i =1,...,p. i 2 i 3.1. DIRECT PRODUCTS, SUMS, AND DIRECT SUMS 153 It is clear that this map is linear, and so its image is a subspace of E denoted by U + + U 1 ··· p and called the sum of the subspaces U1,...,Up. -
Semisimple Categories
Semisimple Categories James Bailie June 26, 2017 1 Preamble: Representation Theory of Categories A natural generalisation of traditional representation theory is to study the representation theory of cate- gories. The idea of representation theory is to take some abstract algebraic object and study it by mapping it into a well known structure. With this mindset, a representation is simply a functor from the category we want to study to a familiar category. For example, consider a group G as a one object category where every morphism is invertible. A rep- resentation of G is typically a group homomorphism from G into End(V ), where V is some vector space. Notice that this is exactly a functor G ! Vec. Now consider a homomorphism between representations. We require such a map to commute with the actions on representations. Such a condition is exactly saying that the homomorphism is a natural transformation between the two representations. Thus, the most general theory of representations amounts to studying functors and natural transforma- tions. Unfortunately, in this general setting, there isn't too much we can say. To `get some more meat on the bones', we might ask that our categories have extra structure. For example, we might want our categories to be monoidal, linear, abelian or semisimple. If you wish, you can justify these extra structures by appealing to the fact that most categories that arise in physics automatically come with them. This report will focus on what we mean by a semisimple category. Note that we have already studied semisimple representations and semisimple algebras (c.f. -
Groups with Preassigned Central and Central Quotient Group*
GROUPS WITH PREASSIGNED CENTRAL AND CENTRAL QUOTIENT GROUP* BY REINHOLD BAER Every group G determines two important structural invariants, namely its central C(G) and its central quotient group Q(G) =G/C(G). Concerning these two invariants the following two problems seem to be most elementary. If A and P are two groups, to find necessary and sufficient conditions for the existence of a group G such that A and C(G), B and Q(G) are isomorphic (existence problem) and to find necessary and sufficient conditions for the existence of an isomorphism between any two groups G' and G" such that the groups A, C(G') and C(G") as well as the groups P, Q(G') and Q(G") are isomorphic (uniqueness problem). This paper presents a solution of the exist- ence problem under the hypothesis that P (the presumptive central quotient group) is a direct product of (a finite or infinite number of finite or infinite) cyclic groups whereas a solution of the uniqueness problem is given only un- der the hypothesis that P is an abelian group with a finite number of genera- tors. There is hardly any previous work concerning these problems. Only those abelian groups with a finite number of generators which are central quotient groups of suitable groups have been characterized before, t 1. Before enunciating our principal results (in §2) some notation and con- cepts concerning abelian groups will have to be recorded for future reference. If G is any abelian group, then the composition of the elements in G is denoted as addition: x+y. -
Group Theory
Algebra Math Notes • Study Guide Group Theory Table of Contents Groups..................................................................................................................................................................... 3 Binary Operations ............................................................................................................................................................. 3 Groups .............................................................................................................................................................................. 3 Examples of Groups ......................................................................................................................................................... 4 Cyclic Groups ................................................................................................................................................................... 5 Homomorphisms and Normal Subgroups ......................................................................................................................... 5 Cosets and Quotient Groups ............................................................................................................................................ 6 Isomorphism Theorems .................................................................................................................................................... 7 Product Groups ............................................................................................................................................................... -
Quotient Spaces and Direct Sums. in What Follows, We Take V As a Finite
Quotient Spaces and Direct Sums. In what follows, we take V as a finite dimensional vector space over a field F . Let W ⊆ V be a subspace. For x and y in V , define x ≡ y mod W if and only if y − x ∈ W It follows that the relation x ≡ y mod W is an equivalence relation on V . Namely (i) x ≡ x mod W for any x ∈ V . (reflexivity) (ii) If x ≡ y mod W then y ≡ x mod W (symmetry) (iii) If x ≡ y mod W and y ≡ z mod W then x ≡ z mod W (transitivity) General theory shows that if we define the equivalence class x = {y|y ≡ x}, then the following properties hold. (iv) The equivalence classes partition V . Namely, distinct equivalent classes are disjoint, and their union is V . (v) x ≡ y mod W if and only if x = y. It is easily shown that x = x + W . Item (v) shows that equivalence can be converted to equality, using equivalence classes. Thus, x = y if and only if x + W = y + W if and only if x ≡ y if and only if y − x ∈ W . The following algebraic properties of equivalence can easily be shown: (vi) If x ≡ y mod W and c ∈ F then cx ≡ cy mod W . (vii) If x ≡ y mod W and u ≡ v mod W then x + u ≡ y + v mod W . Properties (v1) and (vii) can be stated as ”Equivalence is compatible with scalar multiplica- tion and addition.” It allows us to define addition and scalar multiplication of the equivalence classes in the following natural way. -
Section I.8. Direct Products and Direct Sums
I.8. Direct Products and Direct Sums 1 Section I.8. Direct Products and Direct Sums Note. In this section, we introduce general products of groups. This idea will be important when we classify finitely generated abelian groups in Section II.2. Note. We defined the direct product of two groups in Section I.1. We now extend this idea to a collection of groups indexed by an arbitrary index set I (possibly infinite or maybe even uncountable). Definition. Consider an indexed family of groups {Gi | i ∈ I}. Define a binary Q Q operation on the Cartesian product of sets i∈I Gi as follows. If f, g ∈ i∈I Gi (that is, f, g : I → ∪i∈IGi and f(i), g(i) ∈ Gi for all i ∈ I) then fg : I → ∪i∈IGi is Q the function given by i 7→ f(i)g(i) ∈ Gi. So fg ∈ i∈I Gi by Definition 0.5.1. The Q Q set i∈I Gi together with this binary operation (where we identify f ∈ i∈I Gi with its image, the ordered set {ai | i ∈ I}) is the direct product (or complete direct sum) of the family of groups {Gi | i ∈ I}. Q Note. In a crude sense, we can think of the elements of Gi as |I|-tuples in which the binary operation is performed componentwise. If I = {1, 2, . , n} is Q finite, then we denote Gi as G1 × G2 × · · · × Gn (in multiplicative notation) or G1 ⊕ G2 ⊕ · · · ⊕ Gn (in additive notation). I.8. Direct Products and Direct Sums 2 Theorem I.8.1.