Question

Let $H$ be a subgroup of a group$G$ and let $N=\underset{a\in G}{\cap }{a}^{-1}Ha$ .Prove that $N$ is a normal subgroup of $G$.

Short answer

It has been proved that $N$ is a normal subgroup of $G$.

Step-by-step solution

Given that

It is given that $H$ is a subgroup of a group $G$.

Also, $N=\underset{a\in G}{\cap }{a}^{-1}Ha$

It is clear that $N$ is a subgroup of $G$. We need to prove that $N$ is normal in $G$.

By definition, $N\subseteq a^{-1}Ha$for every element $a$.

Prove that N is a normal subgroup of  G

Let $g\in G$.

Therefore, $g^{-1}Ng\subseteq g^{-1}{a}^{-1}Hag={\left(ag\right)}^{-1}H\left(ag\right)$for every $a\in G$.

For any $b\in G$, let $a=b{g}^{-1}$ .

Thus, $g^{-1}Ng\subseteq b^{-1}Hb$

Which implies $g^{-1}Ng\subseteq \underset{b\in G}{\cap }{b}^{-1}Hb$

But $\underset{b\in G}{\cap }{b}^{-1}Hb=N$

Thus $g^{-1}Ng\subseteq N$ for every $g\in G$

Conclusion

Thus it can be concluded that $N$ is a normal subgroup of $G$.