# Structure of composition of permutations

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.

• "if $P_1$ is a simple cycle (that is, a shift with no fixed points), what is the range of $P_1 \cdot P_2$?" - I believe you answered that in your previous paragraph "So their range of composition spans the entire space of permutations" Oct 3, 2021 at 16:34
• @poncho Clarified wording to indicate "what is the range of composition for a known P2" Oct 3, 2021 at 16:56

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.