Writing Dihedral group $D_{8}$ (order 8)as a semidirect product $% V\left\langle \alpha \right\rangle $, where $V$ is a four-group and $\alpha $ an involution, and so we have $D_{8}=\left\langle \alpha ,\beta \right\rangle $ and $V=\left\langle \beta ,\beta ^{\alpha }\right\rangle .$
I want to show that: Let $D$ be isomorphic with $D_{8}=\left\langle \alpha ,\beta \right\rangle $ and suppose that $D$ acts on a group $G$ in such a manner that $C_{G}(V)$ is finte, where $V=\left\langle \beta ,\beta ^{\alpha }\right\rangle $ and $\alpha $ sends every element of $G$ to the inverse, then $C_{G}\left( \beta \right) $ is finite.
My solution:
Let $\phi \in AutG,$ then $g^{\phi \alpha }=\left( g^{-1}\right) ^{\phi }$ and $g^{\alpha \phi }=\left( g^{-1}\right) ^{\phi }$. Therefore $\alpha \in Z\left( AutG\right) $ and so $C_{G}(\left\langle \beta ,\beta ^{\alpha }\right\rangle )=C_{G}\left( \beta \right) $ is finite. ($V=\left\langle \beta \right\rangle$???)
I think there's a problem in my solution for the Klein group is not cyclic.
Help!