  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)}

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

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