Show that \(S_n\) is generated by transpositions.
Solution
Every permutation can be written as a product of disjoint cycles, so it suffices to express a cycle as a product of transpositions.
For a \(k\)-cycle, \[ (a_1a_2\cdots a_k) =(a_1a_k)(a_1a_{k-1})\cdots(a_1a_2). \] Thus every cycle is a product of transpositions, hence every permutation is a product of transpositions.
Therefore \[ S_n=\langle\text{transpositions}\rangle. \] In fact the adjacent transpositions already suffice: \[ S_n=\langle(12),(23),\ldots,(n-1\ n)\rangle, \] because any transposition \((ij)\) can be obtained from adjacent ones.