Constructing Cayley Tables

3 Key excerpts on "Constructing Cayley Tables"

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  Groups, Graphs and Trees

  An Introduction to the Geometry of Infinite Groups

  • John Meier(Author)
  • 2008(Publication Date)
  • Cambridge University Press

    (Publisher)

  1 Cayley’s Theorems As for everything else, so for a mathematical theory: beauty can be perceived but not explained. . –Arthur Cayley An introduction to group theory often begins with a number of examples of finite groups (symmetric, alternating, dihedral, ...) and constructions for combining groups into larger groups (direct products, for exam- ple). Then one encounters Cayley’s Theorem, claiming that every finite group can be viewed as a subgroup of a symmetric group. This chapter begins by recalling Cayley’s Theorem, then establishes notation, termino- logy, and background material, and concludes with the construction and elementary exploration of Cayley graphs. This is the foundation we use throughout the rest of the text where we present a series of variations on Cayley’s original insight that are particularly appropriate for the study of infinite groups. Relative to the rest of the text, this chapter is gentle, and should contain material that is somewhat familiar to the reader. A reader who has not previously studied groups and encountered graphs will find the treatment presented here “brisk.” 1.1 Cayley’s Basic Theorem You probably already have good intuition for what it means for a group to act on a set or geometric object. For example: • The cyclic group of order n – denoted Z n – acts by rotations on a regular n-sided polygon. 1 2 Cayley’s Theorems • The dihedral group of order 2n – denoted D n – also acts on the reg- ular n-sided polygon, where the elements either rotate or reflect the polygon. • We use Sym n to denote the symmetric group of all permutations of [n] = {1, 2, . . . , n}. (More common notations are S n and Σ n .) By its definition, Sym n acts on this set of numbers, as does its index 2 subgroup, the alternating group A n , consisting of the even permutations. • Matrix groups, such GL n (R) (the group of invertible n-by-n matrices with real number entries), act on vector spaces.

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  Modern Algebra

  An Introduction

  • John R. Durbin(Author)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  In proving Cayley’s Theorem, we associate with each element of a group G a permu- tation of the set G. The way in which this is done is suggested by looking at the Cayley table for a finite group. As we observed after Theorem 14.1, each element of a finite group appears exactly once in each row of the Cayley table for the group (if we ignore the row labels at the outside of the table). Thus the elements in each row of the table are merely a permutation of the elements in the first row. What we do is simply associate with each element a of G the permutation whose first row (in two-row form) is the first row of the Cayley table and whose second row is the row labeled by a. If the elements in the first row are a 1 , a 2 , . . . , a n (in that order), then the elements in the row labeled by a will be aa 1 , aa 2 , . . . , aa n (in that order). SECTION 20 CAYLEY’S THEOREM 103 Example 20.1. Consider the Cayley table for Z 6 , given in Example 11.2. The permutation associated with [3] by the idea just described is  [0] [1] [2] [3] [4] [5] [3] [4] [5] [0] [1] [2]  .  Cayley’s Theorem extends this idea to groups that are not necessarily finite, and also establishes that this association of group elements with permutations is an isomorphism. Cayley’s Theorem. Every group is isomorphic to a permutation group on its set of elements. PROOF . Our isomorphism will be a mapping θ : G → Sym(G). We begin by describ- ing the permutation that θ will assign to an element a ∈ G. For a ∈ G, define λ a : G → G by λ a (x ) = ax for each x ∈ G. Each such mapping λ a is one-to-one and onto because each equation ax = b (b ∈ G) has a unique solution in G [Theorem 14.1(c)]. Thus λ a ∈ Sym(G) for each a ∈ G. Now define θ : G → Sym(G) by θ (a) = λ a for each a ∈ G. To prove that θ is one-to- one, suppose that θ (a) = θ (b); we shall deduce that a = b.

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  Mathematics for Physics

  A Guided Tour for Graduate Students

  • Michael Stone, Paul Goldbart(Authors)
  • 2009(Publication Date)
  • Cambridge University Press

    (Publisher)

  14 Groups and group representations Groups usually appear in physics as symmetries of the system or model we are studying. Often the symmetry operation involves a linear transformation, and this naturally leads to the idea of finding sets of matrices having the same multiplication table as the group. These sets are called representations of the group. Given a group, we endeavour to find and classify all possible representations. 14.1 Basic ideas We begin with a rapid review of basic group theory. 14.1.1 Group axioms A group G is a set with a binary operation that assigns to each ordered pair (g 1 , g 2 ) of elements a third element, g 3 , usually written with multiplicative notation as g 3 = g 1 g 2 . The binary operation, or product , obeys the following rules: (i) Associativity: g 1 (g 2 g 3 ) = (g 1 g 2 )g 3 . (ii) Existence of an identity: there is an element 1 e ∈ G such that eg = g for all g ∈ G. (iii) Existence of an inverse: for each g ∈ G there is an element g −1 such that g −1 g = e. From these axioms there follow some conclusions that are so basic that they are often included in the axioms themselves, but since they are not independent, we state them as corollaries. Corollary: (i): gg −1 = e. Proof: Start from g −1 g = e, and multiply on the right by g −1 to get g −1 gg −1 = eg −1 = g −1 , where we have used the left identity property of e at the last step. Now multiply on the left by (g −1 ) −1 , and use associativity to get gg −1 = e. Corollary: (ii): ge = g . Proof: Write ge = g (g −1 g ) = (gg −1 )g = eg = g . Corollary: (iii): The identity e is unique. 1 The symbol “e” is often used for the identity element, from the German Einheit , meaning “unity”. 498 14.1 Basic ideas 499 Proof: Suppose there is another element e 1 such that e 1 g = eg = g . Multiply on the right by g −1 to get e 1 e = e 2 = e, but e 1 e = e 1 , so e 1 = e. Corollary: (iv): The inverse of a given element g is unique.

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