Related to this question and this answer (to a different question) is the following, from Dummit & Foote $\S$ 3.5 # 3.
Prove that $S_n$ is generated by $\left \{ (i \ \ \ i+ 1)| 1 \leq i \leq n - 1 \right \}$ [Consider conjugates, viz. $(2 \ \ 3)(1 \ \ 2)(2 \ \ 3)^{-1}$ ]
The book claims that any permutation is a product of transpositions (without proof, but I can find that through the linked pages), and I think I am supposed to use that, but the hint is confusing me. Isn't $(2 \ \ 3)^{-1}$ just equal to $(2 \ \ 3)$ itself? Here is the proof linked to in the answer above:
Since, for $1 \leq j < k < n$ we have $(j \ \ k + 1) = (k \ \ k+1)(j \ \ k)(k \ \ k+1)$, ...
($S_n = $ the group generated y transposing adjacent points)
Is this sufficient? Don't we need to explicitly look at $\sigma \in S_n$?