In: Advanced Math
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)}
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, .