Frattini subgroup

Summary

The Frattini subgroup $\Phi(G)$ of a group $G$ is the intersection of all maximal subgroups of $G$. It is exactly the set of non-generators: elements that can be dropped from any generating set without changing the group generated. It is a characteristic subgroup, and for a finite group it is nilpotent. For a finite p-group $P$, the quotient $P/\Phi(P)$ is elementary abelian, and its dimension is the minimal number of generators of $P$.