The D-classes have a certain type of hierarchy. And to illustrate this, I'll give relevant properties of the rank of a transformation. The first property is that the rank of a product of two transformations is less than or equal to the minimum of the individual ranks. Let's flesh this out some more. Let's start with what I call the nobility in this hierarchy--the permutations. The noble class would like their wealth and power to stay within their class or family. How do they do that? The idea is to marry someone with the same social class as you are. It's a very common trope in movies and TV. So in the same way, for an "offspring" of a permutation to have the full rank still, both of its "parents" should be permutations. rank(ab)=n if and only if rank(a)=rank(b)=n As part of group theory, we study permutations. This is because a group is embedded in a permutation group by Cayley's theorem. In the finite case, every group of order n is isomorphi...
