Question

Question: If $H$and $K$are subgroups of $G$and $H$is normal in $K$, prove that $K$is a subgroup of $N\left(H\right)$. In other words, $N\left(H\right)$is the largest subgroup of $G$in which $H$is a normal subgroup.

Short answer

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

Step-by-step solution

Given that

Given that H and K are subgroups of G and H is normal in K.

Prove K is subgroup of NH

K is closed under multiplication and inverses, and it contains the identity element.

Therefore, to prove it is subgroup of $N\left(H\right)$-, we just need to show that $K$is contained in $N\left(H\right)$.

Let $k\in K$.

We are assuming that $H$is normal in $K$.

So $k^{-1}Hk=H$, which by definition means that $k\in N\left(H\right)$.

Therefore, $k\underline{\subset }N\left(H\right)$.

Hence proved.