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Surname 1 Student’s Name Instructor’s Name Course Code Date Infinite Symmetric Group and Cayley Graph INTRODUCTION In mathematics, the symmetric group on a set denotes the category comprising all bijections of the collection from the set itself where the function composition represents the group operation. The symmetric group is essential to different areas of mathematics, including invariant theory, the representation of the theory of Lie groups, Galois Theory, and combinatorics (Symmetric Group 1). The symmetric group on a set X describes a group whose basic set is the collection of all bijections from X to X and where the function composition serves as the group operation. The symmetric group of degree n illustrates the symmetric group on the set X as X = 1, 2, 3…., n. The denotation of the symmetric group on a set X exhibits various ways, such as S X , X , Ҩ X , and Sym (X). If X represents set 1, 2, 3…., n, then the symmetric group on is demonstrated as S n , n , Ҩ n , and Sym (n) as well. Symmetric groups are categorized into two, namely the finite and the infinite classes. The behavior of symmetric groups on the two sets is quite different (Symmetric Group 2). Conversely, Carley’s theorem stipulates that every group G is isomorphic to a subgroup of the symmetric group on G. This paper seeks to discuss the infinite symmetric groups and Carley’s graphs, as well as the relationship between the two aspects. INFINITE SYMMETRIC GROUPS Infinite symmetric groups contain few nontrivial normal subgroups, usually exclusively the subgroups of bounded support, and none of the small index. Nevertheless, the standard
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Surname 2 analysis vastly depends on the axiom of choice, though not entirely (McKenzie 1). The construction of infinite symmetric groups is based on the premise that S n acts on X n through the permutation of elements, which include σ (x 1, x 2 …, x n ) = (x σ1, x σ2 …, x σn ) for all σ in S n and (x 1, x 2 …, x n ) in X n . Definition1.1. Let (X, *) represent a pointed topological space. The nth symmetric product of X denotes the orbit space SP n X = X n /S n of the permutation activity for all natural numbers n (Villadsen 2). X n contains the typical product topology whereas the orbit space and the SP n X are given the quotient topology and the base point, (*,*…, *) respectively. This allows the embedment of SP n X into SP n+1 X given by (x 1, x 2 …, x n ) →(x 1, x 2 …, x n , *). Therefore, SP n X is naturally regarded as a subset of SP n+1 X to yield sequence SP1 X SP 2 X ….. SP n X SP n+1 X … that allows the definition of an infinite symmetric product. Definition 1.2 Let (X, *) represent a pointed topological space. The colimit that describes the infinite symmetric product of X is: SP X = colim SP n X. The development of this description this infinite symmetric product involves the orbit space and a colimit.
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  • Spring '16
  • Dr. samuel
  • Symmetric group, Infinite Symmetric Group

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