Has every infinite simple group a faithful irreducible representation?
This question solves the finite case. However, the proof requires a non-trivial linear representation of a finite group. I want to know if the conclusion is true for an infinite simple group. If it is not, can you give me a counterexample?
Best Answer
The statement is true for infinite groups with some exceptions if you allow infinite-dimensional representations
Unless $G$ is a p-group, where $p$ is the characteristic of $k$, the group ring $k[G]$ is not local, so there exist at least two isomorphism classes of simple modules, in particular, there is an irreducible representation where $G$ acts nontrivially.
Such a representation is faithful, because $G$ is simple and so any nontrivial homomorphism from $G$ is injective.
Conversely, if $G$ is $p$-group (such as a Tarski monster), where $p$ is the characteristic of $k$, then $k[G]$ is local and hence there is only one irreducible representation, the trivial one, which is not faithful.