How do I calculate the Frattini subgroup of a simple non-abelian group $G$?
My efforts:
Well, it's a well know fact that $\Phi(G) \trianglelefteq G$ because $\Phi(G)$ is a characteristic subgroup. Thus $\Phi(G) = 1$ or $\Phi(G) = G$ due to the simplicity of $G$.
I'm wondering if the hypothesis '$G$ is non-abelian' helps me decide which of the two possibilities above actually occurs.
Since $G$ is a simple non-abelian group, we obtain $Z(G)=1$ (consequently $G$ is not nilpotent) and $G' = G$. How do I use this to prove what I want?
Remark: Note that the group $G$ is arbitrary, so it can be finite or infinite.