We know that every integer $n\in\Bbb Z$ has an unique prime decomposition $$n=p_1\cdot\dots\cdot p_r$$ where $r$ is number of prime factors $p_i$ of $n$.
We know that every member $\sigma\in S_n$ has an unique cycle decomposition $$\sigma=c_1\cdot\dots\cdot c_r$$ where $r$ is number of cycle factors $c_i$ of $\sigma$.
Can we define addition of $\sigma,\tau\in S_n$?