Mathematics
Group Theory Terminology
Group theory terminology refers to the language used to describe the properties and operations of groups, which are mathematical structures that represent symmetries and transformations. Key terms include group, subgroup, isomorphism, homomorphism, and automorphism, among others. These terms are used to describe the relationships between groups and their elements.
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11 Key excerpts on "Group Theory Terminology"
- Hamid Krim, Abdessamad Ben Hamza(Authors)
- 2015(Publication Date)
- Cambridge University Press(Publisher)
2 Fundamentals of group theory Motivated by the concept of symmetry, we present in this chapter the fundamental elements of group theory. Roughly speaking, symmetry is a transformation that leaves an object of study unchanged, meaning that the object looks the same from different points of view. The set of transformations that characterize the symmetry of an object naturally form a group. Group theory is a branch of mathematics that was inspired by these types of groups, and is of paramount importance in many areas of physics, chemistry, engineering, and computer science. In chemistry, for example, groups are used to classify crystal structures and the symmetries of molecules. In physics, groups are used for solving problems in atomic, molecular, and solid state physics. Groups are extensively used in cryptography, which is the science of encoding information so that only certain specified people can decode it. In addition, group theory has proven very useful in a wide variety of signal and image processing applications, including filter design, image edge detection, and deformable image registration and retrieval. The outline of this chapter is as follows. In Section 2.1, we start by pointing out the interesting connection of groups with symmetry. Just as numbers can be used to measure size, groups can be used to measure symmetry. This relation of groups with symmetry reveals an important linkage between geometry and algebra. Then, we introduce the notion of a group, and describe in detail the group-theoretical concepts through illus- trative examples. In particular, we take a detailed look at subgroups, cosets and normal subgroups, quotient groups, homomorphisms, cyclic and permutation groups. We also highlight matrix groups. Section 2.2 provides a very bare summary of some basic facts about topological spaces and metric spaces.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- White Word Publications(Publisher)
The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—a very active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed conc retely (its group representations), both from a theoretical and a computational point of view. A particularly rich theory has been developed for finite groups, which culminated with the monumental classification of finite simple groups completed in 1983. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become a particularly active area in group theory. Definition and illustration First example: the integers One of the most familiar groups is the set of integers Z which consists of the numbers ..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ... The following properties of integer addition serve as a model for the abstract group axioms given in the definition below. 1. For any two integers a and b , the sum a + b is also an integer. In other words, the process o f adding integers two at a time always yields an integer, not some other type of number such as a fraction. This property is known as closure under addition. 2. For all integers a , b and c , ( a + b ) + c = a + ( b + c ). Expressed in words, adding a to b first, and then adding the result to c gives the same final result as adding a to the sum of b and c , a property known as associativity . ________________________ WORLD TECHNOLOGIES ________________________ 3.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
A starting point is the group Z of integers with addition as group operation, introduced above. If instead of addition multiplication is considered, one obtains multiplicative groups. These groups are predecessors of impor-tant constructions in abstract algebra. Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. For example, Henri Poincaré founded what is now called algebraic topology by introducing the fundamental group. By means of this connection, topological properties such as proximity and continuity translate into properties of groups. i[›] For example, elements of the fundamental group are represented by loops. The second image at the right shows some loops in a plane minus a point. The blue loop is considered null-homotopic (and thus irrelevant), because it can be continuously shrunk to a point. The presence of the hole prevents the orange loop from being shrunk to a point. The fundamental group of the plane with a point deleted turns out to be infinite cyclic, generated by the orange loop (or any other loop winding once around the hole). This way, the fundamental group detects the hole. In more recent applications, the influence has also been reversed to motivate geometric constructions by a group-theoretical background. j[›] In a similar vein, geometric group theory employs geometric concepts, for example in the study of hyperbolic groups. Further branches crucially applying groups include algebraic geometry and number theory. In addition to the above theoretical applications, many practical applications of groups exist. Cryptography relies on the combination of the abstract group theory approach together with algorithmical knowledge obtained in computational group theory, in ________________________ WORLD TECHNOLOGIES ________________________ particular when implemented for finite groups.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- College Publishing House(Publisher)
As these two prototypes are both abelian, so is any cyclic group. The study of abelian groups is quite mature, including the fundamental theorem of finitely generated abelian groups; and reflecting this state of affairs, many group-related notions, such as center and commutator, describe the extent to which a given group is not abelian. Symmetry groups Symmetry groups are groups consisting of symmetries of given mathematical objects—be they of geometric nature, such as the introductory symmetry group of the square, or of algebraic nature, such as polynomial equations and their solutions. Conceptually, group theory can be thought of as the study of symmetry. t[›] Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object X if every group element performs some operation on X compatibly to the group law. In the rightmost example below, an element of order 7 of the (2,3,7) triangle group acts on the tiling by permuting the highlighted warped triangles (and the other ones, too). By a group action, the group pattern is connected to the structure of the object being acted on. Rotations and flips form the symmetry group of a great icosahedron ________________________ WORLD TECHNOLOGIES ________________________ In chemical fields, such as crystallography, space groups and point groups describe molecular symmetries and crystal symmetries. These symmetries underlie the chemical and physical behavior of these systems, and group theory enables simplification of quantum mechanical analysis of these properties. For example, group theory is used to show that optical transitions between certain quantum levels cannot occur simply because of the symmetry of the states involved. Not only are groups useful to assess the implications of symmetries in molecules, but surprisingly they also predict that molecules sometimes can change symmetry.
eBook - PDFAll the Math You Missed
(But Need to Know for Graduate School)
- Thomas A. Garrity(Author)
- 2021(Publication Date)
- Cambridge University Press(Publisher)
11 Algebra Basic Object: Groups and Rings Basic Map: Group and Ring Homomorphisms While current abstract algebra does indeed deserve the adjective abstract , it has both concrete historical roots and modern-day applications. Central to undergraduate abstract algebra is the notion of a group, which is the algebraic interpretation of the geometric idea of symmetry. We can see something of the richness of groups in that there are three distinct areas that gave birth to the correct notion of an abstract group: attempts to find (more accurately, attempts to prove the inability to find) roots of polynomials, the study by chemists of the symmetries of crystals, and the application of symmetry principles to solve differential equations. The inability to generalize the quadratic equation to polynomials of degree greater than or equal to five is at the heart of Galois Theory and involves the understanding of the symmetries of the roots of a polynomial. Symmetries of crystals involve properties of rotations in space. The use of group theory to understand the symmetries underlying a differential equation leads to Lie Theory. In all of these the idea and the applications of a group are critical. 11.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . This section presents the basic definitions and ideas of group theory. Definition 11.1.1 A non-empty set G that has a binary operation G × G → G, denoted for all elements a and b in G by a · b, is a group if the following conditions hold. 1. There is an element e ∈ G such that e · a = a · e = a, for all a in G. (The element e is of course called the identity.) 2. For any a ∈ G, there is an element denoted by a −1 such that aa −1 = a −1 a = e. (Naturally enough, a −1 is called the inverse of a.) 3. For all a,b,c ∈ G, we have (a · b) · c = a · (b · c) (i.e., we must have associativity).
eBook - PDF- Victor P Snaith(Author)
- 1998(Publication Date)
- WSPC(Publisher)
Chapter 1 Group Theory 1.1 The concept of a group The concept of a group, which we are about to study in considerable detail, is one of the many axiomatic structures which constitute the area of abstract algebra. As the name suggests, this collection of mathematical gadgets has arisen in response to the desire to construct algebraic abstractions of familiar phenomena. Although groups are used nowadays in a number of applications ranging from vibrations of chemical molecules to error correcting codes, the fundamental origins of the subject arise from the algebraicisation of the notion of symmetry. The following examples will serve to illustrate what this algebraic abstraction is required and expected to do. Example 1.1.1 Suppose that we are given a rigid three-dimensional solid which we will fondly call X, for brevity. Imagine X firmly implanted stably in some position; for example, X might be a regular tetrahedron resting on a table. A symmetry of X is any operation consisting of picking X up, juggling it around in some manner and then replacing it so as to occupy exactly the same space as before. In abstract algebra it is fashionable (and sensible) to denote things by algebraic symbols; in particular, let us denote the symmetries of X by the symbols si ,S2, — A symmetry of X is not required to return each point of X to the place from which it started. In fact, in order to keep track of what a symmetry of X does, it is a good idea to decorate X in some manner with markers. For example, if X is an equilateral triangle we might number the vertices. Having done this, the six symmetries of the triangle, X y would look as follows: 1 2 CHAPTER 1. GROVP THEORY 1.1.2 Symmetries of an equilateral triangle A AA, A — A A AA, A -,A S A -,A, A -A Question: Why is this list complete? 1.1.3 The symmetries of an equilateral triangle, while only a very simple example, can give us some suggestions of what an algebraic abstraction would be needed for and what it should include.
eBook - PDFMathematical Methods for Physicists
A Concise Introduction
- Tai L. Chow(Author)
- 2000(Publication Date)
- Cambridge University Press(Publisher)
12 Elements of group theory Group theory did not find a use in physics until the advent of modern quantum mechanics in 1925. In recent years group theory has been applied to many branches of physics and physical chemistry, notably to problems of molecules, atoms and atomic nuclei. Mostly recently, group theory has been being applied in the search for a pattern of ‘family’ relationships between elementary particles. Mathematicians are generally more interested in the abstract theory of groups, but the representation theory of groups of direct use in a large variety of physical problems is more useful to physicists. In this chapter, we shall give an elementary introduction to the theory of groups, which will be needed for understanding the representation theory. Definition of a group (group axioms) A group is a set of distinct elements for which a law of ‘combination’ is well defined. Hence, before we give ‘group’ a formal definition, we must first define what kind of ‘elements’ do we mean. Any collection of objects, quantities or operators form a set, and each individual object, quantity or operator is called an element of the set. A group is a set of elements A, B, C ; . . . ; finite or infinite in number, with a rule for combining any two of them to form a ‘product’, subject to the following four conditions: (1) The product of any two group elements must be a group element; that is, if A and B are members of the group, then so is the product AB . (2) The law of composition of the group elements is associative; that is, if A , B , and C are members of the group, then AB C A BC . (3) There exists a unit group element E , called the identity, such that EA AE A for every member of the group. 430 (4) Every element has a unique inverse, A 1 , such that AA 1 A 1 A E . The use of the word ‘product’ in the above definition requires comment.
eBook - PDFPrinciples of Mathematics
A Primer
- Vladimir Lepetic(Author)
- 2015(Publication Date)
- Wiley(Publisher)
5 GROUP THEORY I seem to have been only like a boy playing on the sea shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me. I. Newton 1 5.1 INTRODUCTION There is, arguably, no other mathematical discipline that originated from such simple concepts, and with such profound implications to virtually all branches of mathematics, physics, and many other sciences, as Group Theory. The axiomatics of groups are so simple, so natural, and so “how-could-it-have-been-otherwise,” that one wonders how something so self-evident and so ubiquitous wasn’t dis-covered much earlier in its own right, rather than being motivated by problems in number theory, geometry, the theory of algebraic equations, and the like. And yet, how could something so “simple” could have evolved into such an elabo-rate, encompassing mathematical theory underpinning the fundamental laws of elementary particle physics, the Standard Model, String Theory, and so on. 1 Brewster, D., Life of Sir Isaac Newton, Nabu Press, 2010. Principles of Mathematics: A Primer , First Edition. Vladimir Lepetic. © 2016 John Wiley & Sons, Ltd. Published 2016 by John Wiley & Sons, Ltd. 328 GROUP THEORY It is also safe to say that groups occur abundantly in nature. When looking at natural objects that usually attract attention, say, a beautiful flower, a crys-tal, a geometric figure or object, or even better, a human face, a piece of art, a sculpture, a painting, or still more abstractly, just a few musical tunes from your favorite composition – what makes you say that those “things” are beautiful, har-monious, symmetric; why are they “pleasing to the senses?” Is there something that, regardless of their superficial differences, is common to all of them? It is here that mathematics, abstract algebra in particular, enters the quest. With a lit-tle consideration, one recognizes the structure of a group.
eBook - PDFGroup Theory in Physics
An Introduction to Symmetry Principles, Group Representations, and Special Functions in Classical and Quantum Physics
- Wu-Ki Tung(Author)
- 1985(Publication Date)
- WSPC(Publisher)
CHAPTER 2 BASIC GROUP THEORY Bearing in mind the concrete examples of symmetry groups in physics just described, we shall present in this chapter the key elements of group theory which form the basis of all later applications. In order to convey the simplicity of the basic group structure, we shall stay close to the essentials in the exposition. 2.1 Basic Definitions and Simple Examples Definition 2.1 (A Group): A set {G:a,b,c...} is said to form a group if there is an operation •, called group multiplication, which associates any given (ordered) pair of elements a,beG with a well-defined product a • b which is also an element of G, such that the following conditions are satisfied: (i) The operation • is associative, i.e. a • (b • c) = (a • b) • c for all a,b,ce G; (ii) Among the elements of G, there is an element e, called the identity, which has the property that a • e = a for all a e G; (iii) For each ae G, there is an element a' 1 e G, called the inverse of a, which has the property that a a' 1 — e. All three of the above conditions are essential. From these axioms, one can derive useful, elementary consequences such as: e~ l = e; a' 1 -a = e; and e-a = a, for all ae G. (Notice the order of the elements in comparison to (ii) and (iii) respectively.) The proofs are non-trivial but standard. [See Problem 2.1] For simplicity, the group multiplication symbol • will be omitted whenever no con-fusion is likely to arise (just as in the multiplication of ordinary numbers). Example 1: The simplest group consists of only one element: the identity ele-ment e. The inverse of e is e and the group multiplication rule is ee = e. It is straightforward to see that all the group axioms are satisfied. The number 1 with the usual multiplication constitutes such a group, which we shall denote by C x . Example 2: The next simplest group has two elements, one of which must be the identity; we denote them by {e,a}. According to the properties of e, we must have ee = e and ea = ae = a.
- Zhong-qi Ma, Xiao-yan Gu(Authors)
- 2004(Publication Date)
- World Scientific(Publisher)
Chapter 2 GROUP AND ITS SUBSETS 2.1 Definition of a Group * A group is a set G of elements R satisfying four axioms with respect to the given multiplication rule of elements. The axioms are: a) The set is closed to this multiplication; b) The multiplication between elements satisfy the associative law; c) There is an identity E E G satisfying E R = R; d) The set contains the inverse R-' of any element R E G satisfying R-lR = E. The multiplication rule of elements completely describes the structure and property of a group. A group G is called a finite group if it contains finite number g of elements, and g is called the order of G. Otherwise, the group is called an infinite group. For a finite group, the multiplication rule can be given by the multiplication table, or called the group table. A group is called the Abelian group if the product of its elements is commutable. A few elements in G are called the generators of G if any element in G can be expressed as their product. The rearrangement theorem says RG = GR = G, namely, there are no duplicate elements in each row and in each column of the multiplication table. * Two groups are called isomorphic, G M G', if there is a one-to-one correspondence between elements of two groups in such a way products correspond to products. From the viewpoint of group theory, two isomor- phic groups are the same as each other. 1. Let E be the identity of a group G, R and S be any two elements in the group G, R-' and S-' be the inverses of R and S, respectively. Try to show from the definition of a group: (a) RR-l = E ; (b) RE = R; (c) if T R = R, then T = E; (d) if T R = E , then T = R-'; (e) The inverse of (RS) is S-l R-'. 27 28 Problems and Solutions an Group Theory Solution. The key to the proof is that each element in a group has its inverse. Recall that only the definition of a group and the proved conclusion can be used in the later proof.
eBook - PDF- S. Moran(Author)
- 2000(Publication Date)
- North Holland(Publisher)
CHAPTER 1 SOME NECESSARY GROUP THEORY We s t a r t off with an apparently more general definition of a group. In fact it is completely equivalent to the usual definition. is more convenient from our point of view. However it 1.1. DEFINITION. A group G is a nonempty set S of elements together with an equivalence relation - and a binary operation which is defined for a l l elements x and y of S such that (i) (ii) ( i i i ) (iv) x-y belongs to S for a l l x and y of S; (x*y)*z - x*(y*z) for a l l elements x,y and z of S; there exists an element e of S such that e-x - x for a l l x of S; for every x in S there exists an element x-’-x - e ; i f x - y in S, then Z * X - z*y and X * Z - y-z for a l l z in S. in S such that (v) In the usual definition of a group equality of elements in a s e t is taken to be the equivalence relation - . 1.2. has to add the condition that i f x belongs to H and y - x, then y belongs to H. So H consists of complete equivalence classes. (2) the condition that $ is single-valued as stating that x - y in G1 implies that $(x) - $(y) in G2. NOTE. (1) To the usual definition of a subgroup H of a group G one In the usual definition of homomorphism $ : G1 * G2 one interprets 2 CHAPTER 1 The value of approaching groups in this way is that it makes possiblea more intuitive approach to factor groups. 1.3. subgroup of G. FACTOR GROWS. Let G = (S,-,.) be a group and N be a fixed normal With this data we associate a new group G/N = (S,!,.) which has the same set S and product . but has a more stringent equivalence relation. In fact we define N, by x H y if and only if x-ly belongs to N. I t is now a routine exercise to verify that (a) (b) (c) (S,-,*) satisfies the above axioms (i) - (v) of a group. x - y implies that x k! y for a l l x and y i n G; is an equivalence relation on S; N I t is easy to see that x k! e for a l l x in N and using axiom (v) of a group that x y if and only i f x-ly N, e.
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