A particular generating set of $S_n$

Question

Is the set of all transpositions starting at $1$, i.e. elements of the form $(1,a),a=1,...,n$, a generating set of $S_n$?

By this I know that a generating set of the group $S_n$ is given by the elements of the form $(i,i+1)$. So if I prove that these elements are generated by my elements, it´s over.

Let's consider the permutation $(i,i+1)$.

then: $$\begin{align}(1,i)(1,i+1)(1,i)(1)&=(1,i)(1,i+1)(i)=(1,i)(i)=1\\ (1,i)(1,i+1)(1,i)(i)&=(1,i)(1,i+1)(1)=(1,i)(i+1)=i+1\\ (1,i)(1,i+1)(1,i)(i+1)&=(1,i)(1,i+1)(i+1)=(1,i)(1)=i\end{align}$$ and $$(1,i)(1,i+1)(1,i)(j)=(1,i)(1,i+1)(j)=(1,i)(j)=j$$ for $j\neq i,i+1,1$. So the transpositions starting in $1$ generate $S_n$.

Is this correct?

Answer 1

I'd prove it by noticing $(1,i)(1,j)(1,i)=(i,j)$