All the groups below are supposed finite, and their representations, complex.
An abelian group admits an irreducible faithful representation iff it is cyclic.
A group has all its non-trivial irreducible representations faithful iff it is simple.
Question: What's the classification of the groups admitting an irreducible faithful representation?
Best Answer
This question admits an answer in the following mathoverflow post:
Which finite groups have faithful complex irreducible representations?