Subgroup

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  Abstract 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.

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  Problems And Solutions In Group Theory For Physicists

  • 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.

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  Algebra

  • T T Moh(Author)
  • 1992(Publication Date)
  • WSPC

    (Publisher)

  62 Algebra §3 Subgroups In the preceding section, we study the influence of the transformation group G on the set S, i.e., the set S is separated into many disjoint orbits. In this section, we will study the influence of the set S on the group G. Let us introduce the concept of Subgroups. Definition 2.7. Let (G,*) be a group and H c G be a subset of G. If (H,*) is a group, then it is said to be a Subgroup of (G,*). | Discussion (1) The group G is a Subgroup of G and {e} is a Subgroup, the unit Subgroup of G. Those two Subgroups are called the trivial Subgroups of G. All other Subgroups are called the proper Subgroups of G. (2) Given a non-empty subset H of G, to check if (H,*) is a Subgroup, it is unnecessary to check the associative law, because the law is inherited by H from G. It is necessary and sufficient to prove that for any two elements a, b G H , the element a * 6 _ 1 GH: the condition is certainly necessary. Let b = a. Then e = a * a 1 G H . Let a = e. Then a * ft -1 =e*6~ 1 =6~ 1 G H . Therefore every element b has an inverse 6 1 G H . Replacing b by 6 1 , we have a * (ft 1 ) 1 = a * b G H . Therefore H is closed under *. Therefore (H,*) is a Subgroup. For simplicity, we say H is a Subgroup of G if there is no confusion. | Example 8. All even integers 21 form a Subgroup of (Z, -f )• All odd integers do not form a Subgroup, because the difference of two odd integers is not odd. Let S be any subset of G. Then the subset of all elements of the form Hfinite a » w n e r e either a* G S or a~ l G 5 is a Subgroup of G, tie Subgroup (S) generated by S. For instance 21 = (2) and all multiples of 3 = (3) = 3Z. In general all multiple of n = (n) = nl. Furthermore, it follows from the Euclidean algorithm that all Subgroups of 1 are of the form (n) = nl for some n G Z. | Example 9. (Subgroups of rigid motion groups). We shall use the notations of Example 3.

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  Introduction to Continuous Symmetries

  From Space-Time to Quantum Mechanics

  • Franck Laloë, Nicole Ostrowsky, Daniel Ostrowsky(Authors)
  • 2023(Publication Date)
  • Wiley-VCH

    (Publisher)

  Examples: (i) I , C 3 , C 2 3 all together form a Subgroup of C 3v (isomorphic to Z 3 ), but not just I and C 3 . (ii) R (3) is a Subgroup of the set of displacements in 3-dimensional space. (iii) R (2) is a Subgroup of R (3) . Cayley’s theorem: Any finite group of order N is isomorphic to a Subgroup of S N (group of permu- tations of N objects). This property may be understood by looking at the multiplication table of the group: any element of the group is associated to a row that includes a permutation of the N elements of the group (rearrangement lemma). It is easy to show the isomorphism applying the multiplication law in the initial group and in that of the permutations. 52 A. GENERAL PROPERTIES OF GROUPS Invariant Subgroup: A Subgroup H of G is invariant if: g H g −1 ⊂ H ∀ g ∈ G (II-9a) which means that, for any g belonging to G and h to H, the element: h ′ = g h g −1 (II-9b) also belongs 3 to H. This is equivalent to saying that, for any g ∈ G and h ∈ H, there exists h ′ ∈ H such that: g h = h ′ g (II-10) (or the opposite, interchanging the roles of h and h ′ ). The set of the elements c of G that commute with all the elements of G obviously form an invariant abelian Subgroup, called the center C of the group G. It may happen that C = G (if G is abelian) or, on the contrary, that C consists only of the identity element (example: C 3v ). A group that contains no invariant Subgroup other than the trivial Subgroup (identity element e) is called a “simple group”. A-4. Conjugacy classes Consider two elements g and g ′ of G. g is said to be conjugate of g ′ (through the element f ∈ G) if: g = f −1 g ′ f (II-11) It follows that: • g is conjugate of itself (f = e) • g ′ is conjugate of g (through f −1 ).

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  Mathematical 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 ˆ A BC † . (3) There exists a unit group element E , called the identity, such that EA ˆ AE ˆ A for every member of the group. 430 (4) Every element has a unique inverse, A 1 , such that AA 1 ˆ A 1 A ˆ E . The use of the word ‘product’ in the above definition requires comment.

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  Group 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)

  These are examples of classical groups which occupy a central place in group re-presentation theory and have many applications in various branches of mathe-matics and physics. Clearly, SU(n) and O(n) are Subgroups of U(n) which, in turn, is a Subgroup of GL(n). 2.3 The Rearrangement Lemma and the Symmetric (Permutation) Group The existence of an inverse for every element is a crucial feature of a group. A direct consequence of this property is the rearrangement lemma, which will be used repeatedly in the derivation of important results. Rearrangement Lemma: If p, ft, c e G and pb = pc then b = c. Proof: Multiply both sides of the equation by p 1 . QED This result means: if b and c are distinct elements of G, then pb and pc are also distinct. Therefore, if all the elements of G are arranged in a sequence and are multiplied on the left by a given element p, the resulting sequence is just a rearrangement of the original one. The same, of course, applies to multiplication on the right. Let us consider the case of a finite group of order n. We shall denote elements of the group by {0i,02>--->0n}- Multiplication of each of these elements by a fixed element h results in {hg u hg 2 ,...,hg n } = {g kl9 g h2 ,...,g hn }where{h l9 h 2 ,... 9 h H ) is a permutation of the numbers (l,2,...,n) determined by h. We find, therefore, a natural relationship between a group element h e G and a permutation char-acterized by (h !, h 2 ,..., h n ). x As this correspondence is crucial in group theory, we shall introduce the group of permutations here. An arbitrary permutation of n objects will be denoted by ,_c 2 3 ■ » Pl Pi Ps ' Pn where each entry in the first row is to be replaced by the corresponding one in the second row. The set of n permutations of n objects form a group S„ called the permutation group or the symmetric group. It is not hard to see that one permutation followed by a second results in another permutation.

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