Discriminating and finite iff trivial
The following are equivalent for a group:
(2) implies (1)
This is obvious.
(1) implies (2)
It is easier to prove the result in a somewhat different form: a finite nontrivial group cannot be discriminating.
Suppose is a finite nontrivial group. Suppose is the order of . Consider letters , and the following with :
We note that:
- For any tuple , size considerations force that there exist with such that , so that is trivial.
- However, for any particular word , we can choose a tuple with so that is not universally satisfied (this is where we use nontriviality).
Thus, is not discriminating.