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Abstract AlgebraLet n ≥ 2. Show that Sn is generated by each of the following sets. (a) S1 = {(1, 2), (1, 2, 3), (1, 2, 3, 4), ..., (1, 2, 3,..., n)}(b) S2 = {(1, 2, 3, ..., n-1), (1, 2, 3, ..., n)}

Question

Abstract Algebra

Let n ≥ 2. Show that Sn is generated by each of the following sets. 

(a) S1 = {(1, 2), (1, 2, 3), (1, 2, 3, 4), ..., (1, 2, 3,..., n)}

(b) S2 = {(1, 2, 3, ..., n-1), (1, 2, 3, ..., n)}

check_circleAnswer
Step 1

To prove that the symmetric group Sn is generated by the sets given in (a) and (b)

Step 2

S subset of G generates G if every element of G is a product of finitely many elements (or their inverses) of elements from S.

Let G be any group.
Def intion: S G generates G
if every g e G is a product
g abc with a,b,c..,or
a,b,c1..belong to S
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Image Transcriptionclose

Let G be any group. Def intion: S G generates G if every g e G is a product g abc with a,b,c..,or a,b,c1..belong to S

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Step 3

We need to recall the fact shown. Now, any permutation is a product of cycles and any cycle is a product of transpositions ( a transposition is a permutation of the kind (ij), which interchanges i and j and i...

Basic fact:S, is generated by
the transpositions
(12), (13),.n)
Proof: Any (other) transposition
(j)(li)(1,j 1)
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Basic fact:S, is generated by the transpositions (12), (13),.n) Proof: Any (other) transposition (j)(li)(1,j 1)

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