In all questions suppose $G$ metabelian p-group such that

G is not regular ( so $cl(G) \geq p$ ), G is not a wreath product;

$Z(G) \leq \phi(G)$.

1) Let $M$ normal abelian subgroup of $G$ such that $\frac{G}{M} \cong C_{p^{n}}$ with $n \geq 2$. So it exists an element $g \in G - M$ such that $G=M\langle g\rangle$ and $g^{p^{n}} \in M$. Looking at $G'=[G,G]$, I showed that $G'=[M,g].$ Is it true, under these assumptions, that $C_{G}(G')=MZ(G)$ ?

2) If the answer is no, is there some other assumption for which my thesis is true?

3) In every metabelian p-group G, since G' is abelian, we have that $G' \leq C_{G}(G')$. Are there suitable assumptions for which $G' = C_{G}(G')$? I know that this is true when $G'$ is maximal (but this means G cyclic) and when $G'$ is maximal over normal abelian subgroups.

I edited my post since it was not clear, i apologize for this fact, and i'm grateful for your attention to my problem.

Best regards

Marco, PhD student

state those conditions. Otherwise, you are wasting everyone's time, including yours. Stateexactlywhat you want, with what conditions, rather than pose a question and then complain that the answers are not really what you want. Don't just add it in the comments, edit your question and puteverythingyou want in the question. $\endgroup$ – Arturo Magidin Feb 17 '14 at 15:417more comments