If I know that a certain group has only finitely many irreducible representations (let's say over the complex numbers), is that group necessarily finite?
In the following cases, you can see that there will be an infinite number of irreducible representations(irreps).
- If the group is a direct product of infinitely many (non-trivial)groups.
- If the group is finitely generated, infinite, and abelian.
- If there exists a normal subgroup N such that G/N satisfies conditions 1 or 2.
I'm stuck. Any ideas? Given an arbitrary infinite group, can you construct infinitely many irreps?