# What are the countable groups for which each complex representation is semisimple?

Each complex representation of a finite group is semisimple (i.e. decomposes into a direct sum of irreducible complex representations).

Question: What are the countable groups satisfying the same property?

Remark: We do not restrict to finite dimensional representations.

The characterization is two-way: all $$\mathbb C$$ representations are semisimple iff $$G$$ is finite.