# Show H∩N is a normal subgroup of H

• Jan 30th 2011, 10:50 AM
nikie1o2
Show H∩N is a normal subgroup of H
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.
• Jan 30th 2011, 01:00 PM
DrSteve
It is easy to check that he intersection of 2 subgroups is a subgroup (can you do this?)

Now, let $x\in H\cap N$ and $g\in H$. Then $gxg^{-1}\in H$ because $x$ and $g$ are in $H$ and $H$ is a group. Also, $gxg^{-1}\in N$ because $N$ is normal in $G$.