Question

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

1. H is a normal subgroup of $N\left(H\right)$ .

Short answer

It has been proved that H is a normal subgroup of$N\left(H\right)$ .

Step-by-step solution

Given information

It is given that H is a subgroup of a group 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

If $x\in N\left(H\right)$ , then by definition, $x^{-1}Hx=H$

Thus H is normal in $N\left(H\right)$.

Conclusion

Hence, H is normal in $N\left(H\right)$.