Profinite Frattini subgroup ($\Phi (G)$)

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.