Consider a set $S = \{1,2,\ldots,n\}$, use $\Sigma$ to denote all permutations of the set $S$. The ordered set $\sigma \in \Sigma$ is a permutation of $S$.
Define operators $G_i$ for $i= 1,\ldots, n-1$, such that $G_i(\sigma)$ swaps the the $i$-th number and ($i$+1)-th number in the ordered set $\sigma$.
$\textbf{Problem}:$ Let the ordered set $\sigma_0$ be $\{1,2,\ldots,n\}$, show that any permutation $\sigma \in \Sigma$ can be obtained via applying some sequence of $G_i$ to the set $\sigma_0$.
This is something very simple I encountered in my research. I can think of proving this using the idea of "bubble sort". However, as I an new to the field of combinatorics/group theory, I do not know a good system of notations to present the proof. I searched online and saw the Cauchy's two line notation, but still don't know how to apply it properly. Any help or suggestion regarding the notations will be greatly appreciated.