Every element of $S_n$ can be written as a product of transpositions of the form $(1 i )$, for various $i$

Question

Let $n \geq 2$. Show that every element of $S_n$ can be written as a product of transpositions of the form $(1 i )$, for various $i$.

I have proved by induction that if $n \geq 2$, then every permutation in $S_n$ is a product of transpositions. Using the following, if $\sigma=(123\cdots k ) $ then $(123\cdots k )= (1k)(1(k-1))\cdots(12)$. But I don't know how to relate what I have to this exercise. How could I finish the proof?

Answer 1

Each non-trivial transposition $(a\;b)=(1\;a)(1\;b)(1\;a)$ lies in $\langle\mathcal N\rangle$ where $$\mathcal N=\{(1\;i):1\leq i\leq n\}$$ This demonstrates why $\mathcal S_n=\langle\mathcal N\rangle$ because every permutation is a product of transpositions.