D+F 5.5.1

Let $H,K$ be groups, $\varphi: K \longrightarrow Aut(H)$ and $H,K \leq H \rtimes_\varphi K$.
Prove $C_K(H)=ker\varphi$.

$\textit{Proof}$ : $$k \in{ker\varphi} \Longleftrightarrow \varphi_k=\sigma_{id} \Longleftrightarrow \varphi_k(h)=h \quad \forall h\in{H} \Longleftrightarrow$$ $$k \cdot h = h \Longleftrightarrow khk^{-1} = h \Longleftrightarrow kh=hk \Longleftrightarrow k\in{C_K(H)} \blacksquare$$