Frattini subgroup
= Frattini subgroup
{title2=$\Phi(G)$}
{wiki}
For a <finite group> $G$, the Frattini subgroup $\Phi(G)$ is the intersection of all <maximal subgroups>. Its elements are exactly the <non-generators>: an element in every <maximal subgroup> cannot enlarge a proper generated <subgroup> to $G$, while an element outside a <maximal subgroup> generates $G$ together with it.
For a <finite p-group>,
$$
\Phi(G)=G^p[G,G],
$$
where $G^p$ is generated by all $p$th powers and $[G,G]$ is the <commutator subgroup>. Every <maximal subgroup> has index $p$ and is normal, so its <quotient group> is cyclic of order $p$ and it contains $G^p[G,G]$. Conversely $G/G^p[G,G]$ is an <elementary abelian p-group>; each nonzero <vector> is excluded by some <hyperplane>, whose preimage is a <maximal subgroup>. This proves equality.
Thus $\Phi(C_{p^2})$ is its order-$p$ <subgroup>, whereas $\Phi(C_p\times C_p)=1$. The <Frattini quotient> records the independent generators in the <Burnside basis theorem>.