# Normal closure of a subgroup in another subgroup

Let $G$ be a group with subgroups $H$ and $K$. Is it true that the normal closure of $H$ in $K$, $H^K = \langle aHa^{-1} | a \in K \rangle$, is a subgroup of $K$?

I can see this is true if $H \leq K$ but does it hold in general?

• I can very well imagine it not being true if $G$ is the internal direct product of $H$ and $K$ – tehjh Mar 10 '17 at 10:44

Let $G$ be any abelian group. Then $H^{K}=\langle H\rangle =H$ which is not necessarily a subgroup of $K$.
• Great! it makes sense. Suppose that for your case $G$ is non-abelian and $H^K = H$, would it be correct to say that $K \leq N_G(H)$? – R Maharaj Mar 10 '17 at 10:58