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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.

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Obviously if $S$ generates $G$ then $\overline{S}$ generates $G/\Phi$. On the other hand, if $\overline{S}$ generates $G/\Phi$, then $S$ and $\Phi$ together generate $G$. But $\Phi$ is the set of nongenerators, so ...

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    Ah, I see. Thanks for your help.2012-09-24