Question

Let $H$be a subgr$K\subseteq N\left(H\right)$oup of a group $G$ and let $N\left(H\right)$ be its normalizer (see Exercise 39 in Section 7.3). Prove that

(b) If $H$is a normal subgroup of a subgroup$K$ of $G$, then .

Short answer

It has been proved that $K\subseteq N\left(H\right)$.

Step-by-step solution

Given information

It is given that $H$ is a normal subgroup of a subgroup $K$of $G$and$N\left(H\right)$is its normalizer.

It is known that the normalizer of $H$ is the set $N\left(H\right)=\{x\in G|x^{-1}Hx=H\}$.

Prove that H  is a normal subgroup of  NH

Since $H$ is normal in $K$ this implies for every $x\in K$ , $x^{-1}Hx=H$.

By definition of a normalizer, it is clear that $x\in N\left(H\right)$.

Thus $K\subseteq N\left(H\right)$

Conclusion

Hence,$K\subseteq N\left(H\right)$.