There exists a a metric space such that its group of isometries is isomorphic to $\mathbb{Z}$.
My question is as follows: I have to choose true or false. This is a question from a Graduate school admission test.
There exists a metric space such that its group of isometries is isomorphic to $\mathbb{Z}$.
Is it true?
Solution 1:
HINT:
The solution of @MathematicsStudent1122 is quite salvageable, just consider the space $$\mathbb{Z} \cup (\mathbb{Z}+ \frac{1}{3})\cup (\mathbb{Z}+\frac{1}{2})$$