  n][n]. ForT(o(k)), of T and o. ShowProblem 1: Let n E N and Sn be the set of all bijections r:T, o E Sn denote bythatT O O the composition, k -a) for T σ & S, also ποσ E S,.b) the binary operation o is associativ, i.e., for a, , TE Sn(4 oT) (o ou) oTσοOc) there is a function / E Sn such that for every T E SnLO TTTT = l0 T .Find anexplicit description of : [n] > [n].d) For every o E Sn there is a function TE S withwhere is the bijection in c)e) there is an n E N and o and T withRemark: You just showed that (Sn, o)is a symmetric groupis a non-abelian group. The group Sn

Question help_outlineImage Transcriptionclosen][n]. For T(o(k)), of T and o. Show Problem 1: Let n E N and Sn be the set of all bijections r: T, o E Sn denote by that T O O the composition, k - a) for T σ & S, also ποσ E S,. b) the binary operation o is associativ, i.e., for a, , TE Sn (4 oT) (o ou) oT σο O c) there is a function / E Sn such that for every T E Sn LO TT TT = l0 T . Find an explicit description of : [n] > [n]. d) For every o E Sn there is a function TE S with where is the bijection in c) e) there is an n E N and o and T with Remark: You just showed that (Sn, o) is a symmetric group is a non-abelian group. The group Sn fullscreen
Step 1

The question concerns the group theoretic properties of the symmetric group of order n (the group of permutations of length n).  As per norms, the first three questions are answered

Step 2

Let us give an example of the composition in the case n =3 help_outlineImage TranscriptioncloseS is the set of all bijections T: 1,2,3 ,2,3} that is permtes the set {1,2,3} (one -one onto) fullscreen
Step 3

On this set , we define a muliplication operation , using composition of functions. In this example , the compos... help_outlineImage TranscriptioncloseΕναηple: Let π( 1) - 2, π(2) = 3, π (3) = 1. σ1)-1σ(2) = 3, σ(3) = 2 Thern σπ1)= σ(π(1) = σ(2) = 3 σπ(2)- σ(π(2)) = σ(3) - 2 σπ(3) - σ(π(3)) = σ(1) = 1. Note σπε S, as well fullscreen

Want to see the full answer?

See Solution

Want to see this answer and more?

Our solutions are written by experts, many with advanced degrees, and available 24/7

See Solution
Tagged in

Math 