This is a very nice and non-trivial exercise, and we need to be sure we know/can prove the following:

1) A finite p-group is always nilpotent (hint: if is such a group, then ) ;

2) In a nilpotent group we always have that the Frattini subgroup of (hint: for any

maximal sbgp. is abelian (even cyclic of order a prime) and thus );

3) As , we get that and we're done (hint: the Frattini sbgp. of a

group is the set of all non-generators of the group. Apply now (2))

Tonio

3)