# About commutators and center o a certain group

Let $G$ be a group and $H, K\unlhd\ G$ (normal subgroups), with $K\leq H$ such that $\left[H, Aut(G)\right]\leq K$. Why does this imply that $H/K\leq Z(G/K)$? $Z(G)$ denotes the center of $G$.

• What do you mean by $[H,Aut(G)]$? Where is this commutator taken? – Tobias Kildetoft May 8 '13 at 16:45
• An element in $[H, Aut(G)]$ is described by $h^{-1}h^\alpha$ where $h\in H$, $\alpha\in Aut(G)$ and $h^\alpha$ is the image of $h$ by the automorfism $\alpha$. – graozovsky May 8 '13 at 17:14
• Hint: Consider the automorphism $\varphi_g$ of $G$ that conjugates by some fixed $g\in G$. The fact that $[H,\varphi_g]\leq K$ then tells you what? – Tobias Kildetoft May 8 '13 at 17:23
• I've seen it. Thanks! – graozovsky May 8 '13 at 17:26

We know that for all $\varphi\in\rm{Aut}(G)$ and all $h\in H$ we have $h^{-1}\varphi(h)\in K$. In particular, this holds for the automorphisms $\varphi_g$ given by $\varphi_g(x) = g^{-1}xg$ (for $g\in G$).
Thus, we have that for all $h\in H$ and $g\in G$, $h^{-1}\varphi_g(h) = h^{-1}g^{-1}hg = [h,g]\in K$ which precisely means that $H/K\leq Z(G/K)$.