Let in a finite group. If contains and is a nilpotent group, every Sylow subgroup of has . The Frattini argument gives , and the defining non-generator property of forces . Hence is a nilpotent group.
An element that can always be omitted from a generating family. These are exactly the elements of the Frattini subgroup: a proper generated subgroup is contained in a maximal subgroup, while an element outside a maximal subgroup generates the group together with it.
The Frattini quotient of a finite p-group is an elementary abelian p-group. Its dimension over is the minimum number of generators of ; in particular, it has dimension at least two when is noncyclic.
A subset generates a finite p-group precisely when its images span the Frattini quotient. Indeed implies , since otherwise a maximal subgroup contains both factors. Minimal generating sets correspond to vector-space bases in the quotient.
Articles by others on the same topic
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 \( G \).