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

Abstract Algebra

Let n ≥ 2. Show that S_n 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)}

Solutions

Expert Solution

We know that is generated by the subset of all transpositions in it.

a) Now, observe that Thus, if is the subgroup of generated by then we have

But then, for every integers . That is, contains all the transpositions in . Hence, .

b) Now, observe that . Thus, if is the subgroup of generated by then we have

But then, for every integers , we have

for every integers ; this is because is a transposition as conjugation preserves cycle structure, and sends to . Thus, contains all the transpositions in . Hence, .

Colby Messinger answered 1 year ago

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