I was assigned the following homework problem over break:
Let $g \in S_n$. Show that $g$ can be written as a product of at most $n − 1$ transpositions.
Please do not respond with a proof for this claim, as I would really like to construct it for myself. What I could really use some help understanding is why the following is not a counter example:
Consider the permutation $g=(12)$ in $S_3$. It can easily be checked that $(12)=(12)(12)(12)$, so $g$ can be expressed as the product of 3 transpositions, which is clearly greater than $2=3-1$.
This "counter example" has been bugging me for the past few days. If anyone has any hints on where it fails, I would appreciate that much more than a straight-forward answer.
$(12)$ is the product of just one transposition, thus fewer than $3$.
The fact that $g$ can be expressed as a product of no more than $n-1$ transpositions does not mean it cannot also be expressed as a product of a much larger number of transpositions.