If $P_1, P_2$ are finite permutations, what can we say about $P_3 = P_1 \cdot P_2$? That is, what properties of the composition of permutations can be inferred from the properties of the permutations which are composed?
Since permutations form a group, for any $P_2$ and $P_3$, there exists a $P_1$ that when composed with $P_2$ gives $P_3$. So there range of composition spans the entire space of permutations. That doesn't mean, however, that we can't learn certain things about their structure or nature. For example: If we know the cyclic structure of $P_1$ and $P_2$, can we learn the cyclic structure of $P_3$?
Or, if $P_1$ is a simple cycle (that is, a shift with no fixed points), and $P_2$ is known, what is the range of $P_1 \cdot P_2$?
Or, what is the relationship between $P_1 \cdot P_2$ and $P_2 \cdot P_1$?
More generally: What, if any, properties of composition of permutations can be inferred from the properties of the individual permutations? Or, if you argue that no such properties can be inferred, please prove it.