Typo? "If $G$ is a group, $H$ and $K$ are subgroups of $G$, and $K\unlhd G$. Is $H\cap K \unlhd H$ just if $K \subset H$?"

Question

If $G$ is a group, $H$ and $K$ are subgroups of $G$, and $K\unlhd G$. Is $H\cap K \unlhd H$ just if $K \subset H$?

I believe that my book may have a typo.

Given $x \in H\cap K$, then $x \in H$ and $x \in K$, so, $gxg^{-1} \in K \subset H$

Is this true even when $K \not \subset H$? If yes, I don't know how.

The book is "Abstract Algebra: A First Course" by Dan Saracino.

Answer 1

We have $H\cap K\trianglelefteq H$, with or without such inclusion. Let $h\in H$ and $x\in H\cap K$. Then $hxh^{-1}\in H$ as a product of three elements of $H$, and $hxh^{-1}\in K$ because $K\trianglelefteq G$. So $hxh^{-1}\in H\cap K$.