1. ## Normal Subgroups

Show that if K and N are normal subgroups of G, then K intersect N is normal to G.

We have gkg^-1 is in K and gng^-1 is in N.
I think I know the basic idea:
Take a h in the intersection and take an arbitrary g.
I need to show g^-1hg is in the intersection

I guess I'm stuck in showing that. I guess maybe I don't understand what taking that h means.

2. Originally Posted by kathrynmath
I need to show g^-1hg is in the intersection
Right. Consider $h\in K\cap N$, then,

$h\in K$ and $h\in N$

But $K,N$ are normal subgroups of $G$ ,so:

$ghg^{-1}\in K$ and $ghg^{-1}\in N$ for all $g\in G$ .

which implies $ghg^{-1}\in K\cap N$ for all $g\in G$ .

As a consequence, $K\cap N$ is a normal subgroup of $G$.

Regards.

Fernando Revilla