Let $n\geq 0$ be an integer and $(\tau_{i,i+1})_{1\leq i\leq n-1}$ be a family of transpositions of $\mathfrak{S}_{n}$, the symmetric group of the interval $[1,n]$ of $\mathbb{N}$.
What is the set-theoretic description of the group generated by $(\tau_{i,i+1})_{1\leq i\leq n-1}$? Is it
$$\bigcup_{m\in\mathbb{N}}\{\sigma\in\mathfrak{S}_{n}\ |\ (\exists\alpha)(\alpha\in X^{[1,m]}\land\sigma=\alpha_1...\alpha_m)\},$$ where $X=\{\gamma\ |\ (\exists i)(i\in[1,n-1]\land\gamma=\tau_{i,i+1})\}$?