I am not sure how to do the following question:
Let $G$ be a group and $a \in G$, let $[g,a]=gag^{-1}a^{-1}$, and let $M(a) = \{g \in G | [g,a] \in Z(G) \}$. It is easy to see that the map $\psi : M(a) \rightarrow Z(G)$ defined by $\psi(g) = [g,a]$ is a homomorphism. Does this imply that $M \cong C_G(a) \rtimes Z(G)$ ?
I am not really sure in general how to show that something implies a semidirect product and hence I do not have an idea of how to approach this.
Thank you very much.