 Frattini subgroup

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,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.

The Frattini subgroup is an important concept in group theory, particularly in the study of finite groups. It is defined as the subgroup of a group \( G \) that is generated by all the non-generators of \( G \). Specifically, it has a few equivalent characterizations: 1. **Definition**: The Frattini subgroup \( \Phi(G) \) of a group \( G \) is the intersection of all maximal subgroups of \( G \).