This is a homework question I am having some trouble with.
The claim I'm trying to prove is this: Let $\Phi$ be the Frattini subgroup of a finite group $G$. If $S$ is a subset of $G$, and $\bar{S}$ is the image of $S$ under $G\rightarrow G/\Phi$, then $S$ generates $G$ if and only if $\bar{S}$ generates $G/\Phi$.
I have already shown that $\Phi$ is the set of non-generators of $G$. My inclination is to proceed by cases involving whether or not $S$ is itself contained in a maximal subgroup. Any help would be appreciated.