Let $G$ be the symmetric group $\mathfrak{S}(N)$. Start with the identity member $e$ in $\mathfrak{S}(N)$ and in each step multiply a random transposition. In step $T$, we have a word $$\pi_T = \tau_1 \tau_2\dots \tau_T$$ where $\tau_i$ are transposition. $\pi_T$ has a cycle decomposition \begin{equation*} \sigma_T~=~\sigma_1\sigma_2\ldots, \end{equation*} where $\sigma_i$'s are arranged in decreasing order of length so that $\sigma_1$ (of length $\ell_1$) is the longest cycle. The basic question is, ``What can we say about $\ell_1$ at time $T$? We will study the answer to this question.