# Frattini subgroup contained in center implies derived subgroup is elementary abelian

## Statement

Suppose is a group of prime power order: in other words, has order for some prime and integer . Further, suppose that the Frattini subgroup of is contained in the center of ; in symbols:

Then the commutator subgroup (or derived subgroup) of is elementary Abelian.

## Facts used

- For a group of prime power order, any subgroup containing the Frattini subgroup, has a quotient which is an elementary Abelian group
- For a group of prime power order, the derived subgroup is contained in the Frattini subgroup.
`For full proof, refer: Nilpotent implies derived in Frattini` - In a group of nilpotence class two, the map , for fixed , is an endomorphism.
`For full proof, refer: Class two implies commutator map is endomorphism`

## Proof

, and since , and is Abelian, is Abelian. Hence, to show that it is elementary Abelian, it suffices to show that it is generated by elements of order .

We know that is generated by commutators, i.e. elements of the form where , so it suffices to show that any commutator has order .

Let's do this. Since , the group has nilpotence class two. By fact (2), the map is, for fixed , an endomorphism of . Thus, we have:

contains , so by fact (1), the quotient is elementary Abelian. Equivalently, for any , the element . Thus, the left side in the above equation is the identity element, showing that forany , , completing the proof.