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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
n][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
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n][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

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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

S is the set of all bijections
T: 1,2,3 ,2,3}
that is
permtes the set {1,2,3}
(one -one onto)
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S is the set of all bijections T: 1,2,3 ,2,3} that is permtes the set {1,2,3} (one -one onto)

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Step 3

On this set , we define a muliplication operation , using composition of functions. In this example , the compos...

Εναηple:
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
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Εναηple: 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

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