# The intersection of a normal subgroup and a subgroup is normal in the normal subgroup

Suppose $G$ is a group $H$ is a subgroup, $N$ is a normal subgroup. There is theorem that the intersection of $H \cap N$ is a normal subgroup of $H.$

Is the intersection a normal subgroup of $N?$

Proof? or counter example?

Thanks.

• You mean "we already know the intersection of H and N is a subgroup" instead of "normal"? – Lays Oct 24 '13 at 3:28
• @Lays the intersection is of course a subgroup of N, and there is a theorem that it's a normal subgroup of H. I will mod the question to make it clear. thx – zegautt Oct 24 '13 at 3:33

Pick you favorite group $G$ that has a non-normal subgroup. Mine is probably $S_5$. then if we take $N=S_5$ and $H=\langle (12345) \rangle$ we have a counterexample. Since $N$ is normal in $G$ and $H$ is not normal in $N$ or rather $G$.