# Discriminating and finite iff trivial

From Groupprops

## Statement

The following are equivalent for a group:

- It is both a discriminating group and a finite group.
- It is the trivial group.

## Proof

### (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.