Say G a p-group, M normal abelian subgroup of G such that $\frac{G}{M} \cong C_{p^{n}}.$ 

It exists $g \in G - M $ such that $G=M<g>.$ 


I showed that the derived subgroup is $G' = [M,g].$

1) Is it true that $C_{G}(G')=MZ(G) ?$

2) When is this assertion true?

3) In a metabelian p-group G' is abelian, so it's contained in its centralizer, right? When is it self centralizing? 

Best regards


Marco, PhD student.