Is it true that a compact group always has a faithful, finite-dimensional unitary representation? If not, are there any reasonably simple counter-examples?


I've done some research and know that every group has *some* faithful representation, all irreducible reps of a compact group are finite, and that the irreducible reps separate the points of the group. However, that doesn't quite answer the question!



Finite-dimensional faithful representations of compact groups
A famous theorem is that this is true if and only if $G$ is a Lie group.


