Frattini-in-center p-group implies derived subgroup is elementary abelian
Suppose is a group of prime power order with the property that is elementary Abelian, or equivalently, that is contained in (i.e., is a Frattini-in-center group). Then, the commutator subgroup is an elementary Abelian group: it is an Abelian group of exponent .
- Equivalence of definitions of special group: A special group is a group where the center, commutator subgroup and Frattini subgroup are all equal. In this case, this subgroup is elementary Abelian.
- Frattini-in-center odd-order p-group implies p-power map is endomorphism
- Frattini-in-center odd-order p-group implies (p plus 1)-power map is automorphism
- Commutator subgroup lemma for Frattini-in-center cyclic-center p-group
Given: A finite -group such that is elementary Abelian.
To prove: is elementary Abelian.
Proof: First, since is elementary Abelian, , so is in the center. In particular, is an Abelian group. Thus, it suffices to show that has exponent .
Note that since , has nilpotence class two. Thus, by fact (1), we have that for any , the map is an endomorphism. In particular, we have that for any :
Now, since is elementary Abelian, , so the left side is the identity element. Thus, is the identity element for any , and so is generated by elements of order . Since is Abelian, this tells us that has exponent .