Hierarchy of D-classes determined by rank

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 isomorphic to some subgroup of the permutation group Sn.

In permutation groups, we've explored the interaction between a permutation and another permutation. 

But this time, we will study the interaction of permutations and transformations. Using the social class as a metaphor, we will study the interaction of the nobility and the commoners.

Let's look at two elements from the same D-class. That means then that they have the same rank because a D-class is determined by a rank k. Now if we get the product of two transformation of the same rank, say k, are we assured that the rank of their product is equal to k?

And the answer is no. It may happen, it may not happen.

Having the product of two transformations retain the rank of its "parents" is boring. Let's consider the other case where the rank of the "offspring" is less than the rank of the "parents".

 

 

 

Let k be greater than 1. If b is not a permutation, then there exists an element a in Dk such that rank(ab) is less than or equal to k-1 (downward social mobility of the offspring). In particular, there exists an element a which has the same rank as b but rank(ab) is less than k. That is, their "offspring" has lower rank than them.

Therefore, the rank of a product of two transformations of the same rank is not necessarily preserved.

 

This property is the reason why a transformation which is not a permutation not a part of the separator of the D-class.