What is the symmetry argument here?

Question

Here is a question and answer from "elements of information theory":

What is the "by symmetry" referring to at the end? What is the symmetry argument here?

Answer 1

The "symmetry" refers to the fact that all inputs are equivalent (sort of "locally invariant"), in the sense that each one can transition to the three "next" (cyclically) outputs, and, also, each output can come from three inputs. You should sketch a graph of inputs and outputs with the available transitions if you don't see it.

When one notes this "symmetry" , it should be rather obvious that if all inputs are equiprobable then all outputs are also equiprobable.

The term can also be related with the "symmetric channels" as defined later in the book (7.2).