Structure of composition of permutations

Question

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.

Answer 1

If we know the cyclic structure of $P_1$ and $𝑃_2$, can we learn the cyclic structure of $𝑃_3$?

No. Consider the case when $P_1$ is all fixed points bar a 2-cycle and $P_2$ has the same structure. $P_3$ could be the identity; it could consist of two disjoint 2-cycles and the rest fixed points; it could be a 3-cycle and the rest fixed points. We can say that if $P_1$ and $P_2$ belong to the same subgroup (e.g. membership of the alternating group can be inferred from the cycle structure) then so does $P_3$.

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 $𝑃_1⋅𝑃_2$?

It's the union of right cosets of shift subgroups whose intersection is $P_2$. This is close to tautologous, but I can't think of a better way to describe it.

what is the relationship between $𝑃_1⋅𝑃_2$ and $𝑃_2⋅𝑃_1$?

It's a conjugate by $P_2$ (and so in particular has the same cycle structure). Let $Q=P_1P_2$ so that $P_1=QP_2^{-1}$ and $P_2\cdot P_1=P_2QP_2^{-1}$. $Q$ has $n!$ such representations there is a representation corresponding to any given conjugate and so no further structure can be inferred.