Problem: Consider the arbitrary 2-cycle $(a\ b)$ from $S_n$. Find a way to write this permutation as a product of adjacent 2-cycles.
What I do know:
A transposition is a single cycle of length 2. An adjacent transposition is of the form $(i\ i+1)$. For example, $(3\ 7)$ is a non-adjacent transposition, but $(3,4)$ is an adjacent transposition. It turns out that the set of transpositions for $S_n$ is a generating set for $S_n$.
So, to write the arbitrary 2-cycle $(a\ b)$ from $S_n$ as a product of 2 adjacent cycles, would it look something like this:
I can start at some arbitrary values, say, $(1\ 2)$. Then,
$(1\ 2)(a\ a+1)(a+1\ a+2)\cdots(b-2\ b-1)(b-1\ b)$ and eventually, I will get to $(a\ b)$?
I guess my question is, would this be a valid answer or would I need to generalize it more? And, is it ok to start at some arbitrary values such as the one I chose? Thanks for your help.