Suppose we have that the trivial representation is the only irreducible representation. I am trying to understand what this would imply for any given representation $(V, \phi).$ Couple of questions:
If $V$ is a degree 1 representation does this mean that $V$ is the trivial representation? Clearly this should be true as $V$ has no proper subspaces but is this true in general? I.e., are there degree one representations that are not trivial?
Suppose $V$ is of degree $n > 1.$ Does this mean that we can reduce $V$ to $V_1 \oplus \ldots \oplus V_n$ such that $V_i = ke_i$ is of degree one and that for every $g \in G, ge_i = e_i?$