Every elementary abelian p-group occurs as the Frattini quotient of a p-group in which every maximal subgroup is characteristic
Suppose is a prime number and is an elementary Abelian -group (equivalently, is a vector space over the prime field . Then, there exists a finite -group such that (where denotes the Frattini subgroup of ) and such that every maximal subgroup of is a characteristic subgroup.
For the case that is one-dimensional, we can take . For the case that , apply fact (1) with the subgroup as the trivial group.