Hi everyone,
the question is:
If N is normal subgroup of G and H is any subgroup of G. Show H∩N is a normal subgroup of H.
It is easy to check that he intersection of 2 subgroups is a subgroup (can you do this?)
Now, let $\displaystyle x\in H\cap N$ and $\displaystyle g\in H$. Then $\displaystyle gxg^{-1}\in H$ because $\displaystyle x$ and $\displaystyle g$ are in $\displaystyle H$ and $\displaystyle H$ is a group. Also, $\displaystyle gxg^{-1}\in N$ because $\displaystyle N$ is normal in $\displaystyle G$.