If $K$ and $H$ are subgroups of $G$ and $H \triangleleft K$ then $K \subseteq N(H)$.

Question

We can easily prove the truth of this statement. My question is that why do we not simply say $K=N(H)$ $?$

I'd be really grateful for an elaboration on this.

Answer 1

A simple example of proper containment is when $H$ is not normal, has a nontrivial normalizer, and $H=K$. In general any proper subgroup of the normalizer that contains $H$ will serve as an example.

Answer 2

You might want to specify over which group you take the normalizer; I think this will clear the confusion. While it is true that $H\triangleleft K\Rightarrow K = N_K(H)$, it will not always be that $K = N_G(H)$. However, it is true that $K \subseteq N_G(H)$.

For an example of the strict inclusion, take $G =\mathfrak S_4$, $H = \langle(1\,2\,3)\rangle$. Then $K = \langle (1\,2),(1\,2\,3)\rangle\subsetneq N_G(H)$.