Subgroups and Intersection of Normal Subgroups

Quote:

Originally Posted by kathrynmath

Prove that if N is a normal subgroup of G and H is any subgroup of G, then H intersect N is a normal subgroup of H.

N is normal so gng^-1 is in N for all n in N and g in G.
H is a subgroup so H is closed, has an identity, and has an inverse.

I don't know how to apply these definitions when thinking about the intersection

Let $h\in H$ be arbitrary. Note first that $h\left(N\cap H\right)=(hN)\cap (hH)$ . To see this merely note that $x\in h\left(N\cap H\right)\Leftrightarrow x=hy\text{ }y\in H\cap N\Leftrightarrow x=hy\text{ }y\in N\text{ and }x=hy\text{ }y\in H\Leftrightarrow x\in hN\text{ and }x\in hH$ . Similarly, $\left(N\cap H\right)h=(Nh)\cap (Hh)$ .

Thus, we note that $h\left(N\cap H\right)=(hN)\cap (hH)=(hN)\cap H=(Nh)\cap H=(Nh)\cap (Hh)=(N\cap H)h$ .