  In this paper, we present a detailed proof for the exhibition of a cutoff for
the one-sided transposition (OST) shuffle on the generalized symmetric group
$G_{m,n}$. Our work shows that based on techniques for $m \leq 2$ proven by
Matheau-Raven, we can prove the cutoff in total variation distance and
separation distance for an unbiased OST shuffle on $G_{m,n}$ for any fixed $m
\geq 1$ in time $n \log(n)$. We also prove the branching rules for the simple
modules of $G_{m,n}$ and lay down some of the mathematical foundation for
proving the conjecture for the cutoff in total variation distance for any
general biased OST shuffle on $G_{m,n}$.
