= Profinite Frattini subgroup
{title2=$\Phi(G)$}
The intersection of the maximal proper open <subgroups> of a <profinite group>. A subset topologically generates the <profinite group> exactly when its image topologically generates its quotient by this intersection: a proper closed generated <subgroup> is contained in a maximal open <subgroup>, obtained from a suitable finite quotient. For a <pro-p group>, $\Phi(G)=\overline{G^p[G,G]}$, where the bar denotes <closure> of the algebraically generated power and <commutator subgroups>. Closedness cannot simply be discarded in infinite rank.
Back to article page
