Mathematics
Group Mathematics
Group Mathematics is a branch of abstract algebra that studies the properties of groups, which are sets of elements that can be combined using a binary operation. It explores the structure and symmetry of groups, and their applications in various fields such as physics, chemistry, and computer science.
Written by Perlego with AI-assistance
Related key terms
1 of 5
8 Key excerpts on "Group Mathematics"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- College Publishing House(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 2 Group (Mathematics) The possible manipulations of this Rubik's Cube form a group In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity and invertibility. Many familiar mathematical structures such as number systems obey these axioms: for example, the integers endowed ________________________ WORLD TECHNOLOGIES ________________________ with the addition operation form a group. However, the abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way, while retaining their essential structural aspects. The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics. Groups share a fundamental kinship with the notion of symmetry. A symmetry group encodes symmetry features of a geometrical object: it consists of the set of transfor-mations that leave the object unchanged, and the operation of combining two such transformations by performing one after the other. Such symmetry groups, particularly the continuous Lie groups, play an important role in many academic disciplines. Matrix groups, for example, can be used to understand fundamental physical laws underlying special relativity and symmetry phenomena in molecular chemistry. 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.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- White Word Publications(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 2 Group (Mathematics) The possible manipulations of this Rubik's Cube form a group. In mathematics, a group is an algebraic structure consisting of a set together with an operation that combin es any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity and invertibility. Many familiar mathematical structures such as number systems obey these axioms: for example, the integers endowed with the addition operation form a group. However, the abstract formalization of the ________________________ WORLD TECHNOLOGIES ________________________ group axioms, detached as it is from the concrete nature of any particular group and its operation, al lows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way, while retaining their essential structural aspects. The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics. Groups share a fundamental kinship with the notion of symmetry. A symmetry group encodes symmetry features of a geometrical object: it consists of the set of transformations that leave the object un changed, and the operation of combining two such transformations by performing one after the other. Such symmetry groups, particularly the continuous Lie groups, play an important role in many academic disciplines. Matrix groups, for example, can be used to understand fundamental physical laws underlying special relativity and symmetry phenomena in molecular chemistry. 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.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 2 Group (Mathematics) The possible manipulations of this Rubik's Cube form a group. In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity and invertibility. Many familiar mathematical structures such as number systems obey these axioms: for example, the integers endowed ________________________ WORLD TECHNOLOGIES ________________________ with the addition operation form a group. However, the abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way, while retaining their essential structural aspects. The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics. Groups share a fundamental kinship with the notion of symmetry. A symmetry group encodes symmetry features of a geometrical object: it consists of the set of trans-formations that leave the object unchanged, and the operation of combining two such transformations by performing one after the other. Such symmetry groups, particularly the continuous Lie groups, play an important role in many academic disciplines. Matrix groups, for example, can be used to understand fundamental physical laws underlying special relativity and symmetry phenomena in molecular chemistry. 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.
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 - PDFAbstract Algebra
A Gentle Introduction
- Gary L. Mullen, James A. Sellers(Authors)
- 2016(Publication Date)
- CRC Press(Publisher)
Chapter 2 Groups 2.1 Definition of a group ............................................. 33 2.2 Examples of groups .............................................. 34 2.3 Subgroups ........................................................ 43 2.4 Cosets and Lagrange’s Theorem ................................. 49 In this chapter we discuss the concept of a group. This is one of the most fundamental concepts in abstract algebra. Many other algebraic structures contain groups. In subsequent chapters we will see that every ring contains a group and every field (including the real numbers) contains two groups. Vector spaces are examples of algebraic objects that contain a linear structure, and they also always contain a group. Rings, fields, and vector spaces will be studied in detail in Chapters 3, 4, and 6. It turns out that many algebraic objects with which you are already famil-iar actually form groups. After looking at numerous examples, we will discuss a variety of properties satisfied by groups. 2.1 Definition of a group Definition 2.1 A group is a non-empty set G with a binary operation ∗ such that the following properties hold: 1. Closure: a ∗ b ∈ G for each a, b ∈ G ; 2. Associativity: ( a ∗ b ) ∗ c = a ∗ ( b ∗ c ) for all a, b, c ∈ G ; 3. Identity: There is an element e ∈ G, called the identity element of G, such that e ∗ a = e = a for all a ∈ G ; a ∗ 4. Inverses: For each a ∈ G there is an element a − 1 ∈ G , called the inverse of a , such that a ∗ a − 1 = a − 1 a = e . ∗ We will often abuse the notation above and refer to the group G (rather than the set G ), and suppress the operation ∗ and simply write ab instead of a ∗ b . 33 34 Abstract Algebra: A Gentle Introduction A group G is Abelian or commutative if a ∗ b = a for all a, b ∈ G . b ∗ Note that being Abelian is not part of the definition of a group. As we will see, some groups are Abelian and some are not.
eBook - PDFPrinciples of Mathematics
A Primer
- Vladimir Lepetic(Author)
- 2015(Publication Date)
- Wiley(Publisher)
With a lit-tle consideration, one recognizes the structure of a group. Take an object, any object X , endowed with some structure and consider a map ς ∶ X → X that maps the object onto itself while preserving that structure – we say that there is a sym-metry. Group theory formalizes the essential aspects of symmetry – we say that symmetries form a group. By focusing on the very transformations themselves, we gain an insight into the fundamental makeup of the object we are studying. 5.2 FUNDAMENTAL CONCEPTS OF GROUP THEORY Definition 5.1 Given a set G , by a binary operation “ ∗ ” on G , we mean a function ∗∶ G × G → G that maps any ordered pair of elements of G to an ele-ment of G . In particular, if an ordered pair ( a , b ) ∈ G is mapped into an element c ∈ G , then we write c = a ∗ b . The operation “ ∗ ” could be our usual addition “ + ,” or multiplication “ ⋅ ,” per-formed on a set of real numbers or functions, but it could also be something more abstract, say, permutations, rotations, or translations, and so on, performed on sets that are definitely not sets of numbers. Definition 5.2 A binary operation “ ∗ ” is said to be well defined on a set X if (i) Exactly one element is assigned to each possible ordered pair of elements of X . (ii) For each ordered pair of elements of X , the element assigned to it is again an element of X We say that X is closed under “ ∗ ,” or that closure is satisfied. Example 5.1 Suppose we take the set Z and consider the usual addition “ + ,” or multiplication “ ⋅ ,” as the binary operation “ ∗ ,” such that an ordered pair of two elements of Z , say, ( 2 , 3 ) is mapped into 2 + 3 = 5 by “ + ,” or into 2 ⋅ 3 = 6 by “ ⋅ .” Note that the “results” of our “operations” are again elements of Z . We say that our operations are well defined since they satisfy both conditions (i) and (ii) in Definition 5.2. ◾
eBook - PDFMath Unlimited
Essays in Mathematics
- R. Sujatha, H. N. Ramaswamy, C. S. Yogananda(Authors)
- 2011(Publication Date)
- CRC Press(Publisher)
Part I Mathematics for its Own Sake This page intentionally left blank This page intentionally left blank C hapter 1 Group Theory— What’s Beyond B. Sury Stat-Math Unit, Indian Statistical Institute, 8th Mile Mysore Road, Bangalore 560059, India. e-mail: [email protected] Birth and infancy of group theory Major progress in group theory occurred in the nineteenth century but the evolution of group theory began already in the latter part of 18th century. Some characteristic features of 19th century mathematics which had cru-cial impact on this evolution are concern for rigor and abstraction and the view that mathematics is a human activity without necessarily referring to physical situations. In 1770, Joseph Louis Lagrange (1736–1813) wrote his seminal memoir Reflections on the solution of algebraic equations . He considered ‘abstract’ questions like whether every equation has a root and, if so, how many were real / complex / positive / negative? The problem of al-gebraically solving 5th degree equations was a major preoccupation (right from the 17th century) to which Lagrange lent his major e ff orts in this pa-per. His beautiful idea (now going under the name of Lagrange’s resolvent) is to ‘reduce’ a general equation to auxiliary (resolvent) equations which have one degree less. Later, the theory of finite abelian groups evolved from Carl Friedrich Gauss’s famous “Disquisitiones Arithmeticae”. Gauss (1777–1855) established many of the important properties though he did not use the terminology of group theory. In his work, finite abelian groups 4 Math Unlimited appeared in di ff erent forms like the additive group Z n of integers mod-ulo n , the multiplicative group Z ∗ n of integers modulo n relatively prime to n , the group of equivalence classes of binary quadratic forms, and the group of n -th roots of unity. In 1872, Felix Klein delivered a famous lec-ture A Comparative Review of Recent Researches in Geometry .
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
