I am going through a proof of the following result from my lecture notes: Suppose $\mathbb C G \cong \bigoplus_{i=1}^nU_i$ where the $U_i$ are simple $\mathbb CG $ modules. Then any simple $\mathbb CG$ module $S$ has $S \cong U_i$ for some $i$.
The first line of the proof says:
Let $s \in S \setminus \{0\}$. Then $f:\mathbb CG \to S$ given by $f(a)=a\cdot s$ is surjective.
I can't see why this is necessarily true. I know that it suffices to prove that any two non-zero elements of $S$, $s$ and $t$ have $t=a\cdot s$ for some $a \in \mathbb CG$. This seems true intuitively, but I don't know how to prove it.
Note that $S$ is simple. Now consider $U := {\rm im}\, f$. $U$ is a submodule of $S$, which contains $s$ (as $s = 1 \cdot s = f(1)$). Hence, $U \ne \{0\}$. As $S$ is simple, $U = S$. So $f$ is onto.