Originally Posted by

**tonio** 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 $\displaystyle G$ is such a group, then $\displaystyle |Z(G)|>1$) ;

2) In a nilpotent group $\displaystyle G$ we always have that $\displaystyle G'<Frat(G)=$ the Frattini subgroup of $\displaystyle G$ (hint: for any

maximal sbgp. $\displaystyle M\leq G\, , \, G/M$ is abelian (even cyclic of order a prime) and thus $\displaystyle G'\leq M$);

3) As $\displaystyle G/G'=<xG'>$ , we get that $\displaystyle G=<x>G'\Longrightarrow G=<x>$ 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)