Mathematics
Group Generators
Group generators are elements of a group that can be used to generate the entire group by combining them in various ways. They are often used to study the structure of groups and to prove properties of groups. The number of generators needed to generate a group is called the rank of the group.
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7 Key excerpts on "Group Generators"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
In multiplicative notation, the elements of the group are: ..., a −3 , a −2 , a −1 , a 0 = e , a , a 2 , a 3 , ..., where a 2 means a • a , and a −3 stands for a −1 • a −1 • a −1 =( a • a • a ) −1 etc. h[›] Such an element a is called a generator or a primitive element of the group. A typical example for this class of groups is the group of n -th complex roots of unity, given by complex numbers z satisfying z n = 1 (and whose operation is multiplication). Any cyclic group with n elements is isomorphic to this group. Using some field theory, the group F p × can be shown to be cyclic: for example, if p = 5, 3 is a generator since 3 1 = 3, 3 2 = 9 ≡ 4, 3 3 ≡ 2, and 3 4 ≡ 1. Some cyclic groups have an infinite number of elements. In these groups, for every non-zero element a , all the powers of a are distinct; despite the name cyclic group, the powers of the elements do not cycle. An infinite cyclic group is isomorphic to ( Z , +), the ________________________ WORLD TECHNOLOGIES ________________________ group of integers under addition introduced above. 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.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- College Publishing House(Publisher)
If instead of addition multiplication is considered, one obtains multiplicative groups. These groups are predecessors of important 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 ________________________ WORLD TECHNOLOGIES ________________________ 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 particular when implemented for finite groups.
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.
eBook - PDFGroup Theory for Physicists
With Applications
- Pichai Ramadevi, Varun Dubey(Authors)
- 2019(Publication Date)
- Cambridge University Press(Publisher)
1 Introduction We observe objects in nature to have some pattern or symmetry. In fact, the symmetries inherent in any physical system play a very crucial role in the study of such systems. Group theory is a branch of mathematics that facilitates classification of these symmetries. Hence learning the group theory tools will prove useful to studying applications in physics. Readers will particularly appreciate the power and elegance of the group theoretical techniques in reproducing the experimental observations. Before delving into its applications, it is important to understand the concept of an abstract group from a purely mathematical standpoint. In this chapter, we present the formal definition of a group and also the notations which will be followed in the rest of the book. 1.1 Definition of a Group Definition 1. A group G is a set on which is defined a binary operation called the product having the following properties: 1. Closure: For all a and b in G, the product ab is in G. Here a and b need not be distinct. 2. Associativity: ( ab)c = a(bc) for all a, b and c in G. 3. Existence of Identity: There exists a unique e in G such that ae = ea = a for all a in G. e is called the identity element of the group. 4. Existence of Inverse: For every a in G there exists a unique b in G such that ab = ba = e. b is called the inverse of a and is conventionally denoted by a -1 . 2 Group Theory for Physicists In addition to the above mentioned axioms, if it is also true that ab = ba for all a and b in G, then G is said to be an abelian group. It must be noted that the property of being abelian is special in the sense that not all groups need be abelian. In any group, it is trivial to prove the following statements: ax = bx ⇒ a = b (1.1.1) xa = xb ⇒ a = b ( ab) -1 = b -1 a -1 . Due to the obvious simplicity of the definition, many familiar sets in mathematics are indeed seen to be examples of groups.
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 - PDF- D. J. H. Garling(Author)
- 2021(Publication Date)
- Cambridge University Press(Publisher)
PART I The Algebraic Background 1 Groups It is likely that the reader has already met the concept of a group. It was Galois who first understood the imporance of groups in the study of the roots of a polynomial equation; since then, group theory has blossomed, and developed as a subject in its own right. In this chapter we simply develop those parts of the theory which we shall need later; one of the main purposes is to explain the notation and terminology that we shall use. 1.1 Groups Suppose that S is a set. A law of composition ◦ on S is a mapping from the Cartesian product S ×S into S ; that is, for each ordered pair (s 1 ,s 2 ) of elements of S there is defined an element s 1 ◦ s 2 of S . A group G is a non-empty set with a law of composition ◦ : G × G → G with the following properties: (i) g 1 ◦ (g 2 ◦ g 3 ) = (g 1 ◦ g 2 ) ◦ g 3 for all g 1 ,g 2 ,g 3 in G – that is, composition is associative; (ii) there is an element e in G (the unit or neutral element) such that e ◦ g = g ◦ e = g for each g in G; (iii) to each g in G there corresponds an element g −1 (the inverse of g) such that g ◦ g −1 = g −1 ◦ g = e. Exercise 1.1 Suppose that G is a group. Show that the identity element e is unique, and that for each g ∈ G the inverse element g −1 is also unique. 3 4 1 Groups Two elements g and h of a group commute if g ◦ h = h ◦ g. The commutator [g,h] of g and h is the element g −1 ◦ h −1 ◦ g ◦ h; thus g and h commute if and only if [g,h] = e. A subset A of a group G is said to be commutative, or abelian, if and only if any two elements of A commute. The notation that is used for the law of composition varies from situation to situation. Frequently, there is no symbol, and elements are simply placed side by side: g ◦ h = gh. When G is abelian, it often happens that the law is denoted by g ◦ h = g + h, the identity element is denoted by 0 and the inverse of an element g is denoted by −g.
eBook - PDF- Willard Miller(Author)
- 1973(Publication Date)
- Academic Press(Publisher)
Chapter 1 Elementary Group Theory 1.1 Abstract Groups A group is an abstract mathematical entity which expresses the intuitive concept of symmetry. Defintion. A group G is a set of objects (8, / I , k , . . .I (not necessarily count- able) together with a binary operation which associates with any ordered pair of elements g, Ii in G a third element gli. The binary operation (called group multiplication) is subject to the following requirements: ( I ) There exists an element e in G called the identity element such that ge - eg = g for all g F- G. (2) For every g c G there exists in G an inverse element g-' such that (3) Associative law. The identity (gh)k = g(hk) is satisfied for all gg-1 : g-lg = e. g, 11, k E G. Thus, any set together with a binary operation which satisfies conditions (1)-(3) is called a group. If gk hg we say that the elements g and /I commute. If all elements of G commute then G is a commutative or abelian group. If G has a finite number of elements it has finite order n(G), where n(G) is the number of elements. Otherwise. G has infinite order. A subgroup H of G is a subset which is itself a group under the group multiplication defined in G. The subgroups G and {e} are called improper subgroups of G. All other subgroups are proper. 1 2 1 ELEMENTARY GROUP THEORY Theorem 1.1. A nonempty subset H of a group G is a subgroup if and only if the following two conditions hold: (1) If h, k E H then hk t H . ( 2 ) If h E H then h -* E H . Proof. If H is a subgroup then (1) and (2) clearly hold. Conversely, suppose these conditions hold. We show that Hsatisfies the requirements for a group. The associative law holds since it holds for G. There exists some h E H since H is nonempty and by ( 2 ) we have h-' E H. By (I), Ah-' = e t H, so H has an identity. Q.E.D. The identity element e of a group is unique: Suppose e' E G such that e'g = ge' = g for all g E G. Setting g = e, we find ee' = e'e = e. But e'e = e' since e is an identity element.
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