Nilpotent implies derived in Frattini
Further information: commutator subgroup
The commutator subgroup of a group is defined as the intersection of all its Abelian-quotient subgroups, i.e. normal subgroups whose quotient group is Abelian.
Further information: Frattini subgroup
The Frattini subgroup of a group is defined as the intersection of all its maximal subgroups.
- Nilpotent implies every maximal subgroup is normal: In a nilpotent group, every maximal subgroup is normal, and hence, a maximal normal subgroup
- Any quotient of a nilpotent group is nilpotent. In particular, any quotient by a maximal normal subgroup is a simple nilpotent group
- Any simple nilpotent group is Abelian
Given: A nilpotent group , with derived subgroup and Frattini subgroup
Proof: Since is the intersection of all normal subgroups with Abelian quotients, and is the intersection of all maximal subgroups, it suffices to show that any maximal subgroup is normal with Abelian quotient.
By fact (1), every maximal subgroup is normal, and hence maximal normal. By fact (2), the quotient is a simple nilpotent group. By fact (3), the quotient is forced to be an Abelian group. Thus, every maximal subgroup is normal with Abelian quotient, and we are done.
The result holds in a somewhat greater generality than nilpotent groups, if we can guarantee the following two things:
- Any maximal subgroup is normal
- Any simple quotient is Abelian