If $C_{G}H = G$, then $H \leq Z(G)$.

Question

I at a point in a proof where I should have this statement be true: If $C_{G}(H) = G$, then $H \leq Z(G)$.

I don't know why, but I am completely drawing a blank as to why this assumption is true.

I know that $C_{G}(H) = \{ g \in G | ghg^{-1}=h$ for all $h \in H \} = G$ and that $Z(G) = \{ g \in G |gx=xg for all x \in G \} = \{ g \in G |gxg^{-1}=x$ for all $x \in G \}$. So I can see that the centralizer and the center are looking very similar. But I don't know why $H \leq Z(G)$.

Please help.

Answer 1

Maybe you should try to prove first that $H \subset Z(G)$.

If $h \in H$, since $C_G(H) = G$, for all $x\in G$ we have $hx = xh$, that is, $h \in Z(G)$