Proving that subgroup of Rotations in $D_{2n}$ is a normal subgroup

Question

I am trying to prove that the group of rotations in $D_{2n}$ is a normal subgroup.

I know this group has a cardinality $n$, and the ratio of the $D_{2n}$ and $R$ is $2.$

I am trying to construct a homomorphism that sends $<R>$, generator group of rotations to an identity element...

Any clues?

Answer 1

Well, as you've noted, $|D_{2n}|/|K|=2$, and so $[D_{2n}:K]=2$ which implies $K$ is normal: see Subgroup of index 2 is Normal.