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 $G=M\langle g\rangle$.
I showed that the derived subgroup is $G' = [M,g]$.
Is it true that $C_{G}(G')=MZ(G)$ ?
When is this assertion true?
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.
