Answer to Question #203320 in Abstract Algebra for Raghav

Question

Check whether the subgroup of reflections and subgroup of rotations in D_2n is normal in D_2n or not. (Note that D_2n is the group of symmetries of an n-gon.)

Answer 1

It should be obvious that the rotations form a normal subgroup of index 2: the fact that it is of index 2 is sufficient to prove normality.

Subgroups containing only reflections and the identity must have order a power of 2. Since a reflection times a reflection is a rotation, with $(r^ks)(r^ms)=r^{k−m},$ it should be clear that any such subgroup is in fact of order 2 (we must have k = m). These subgroups are typically NOT normal, but there is an exception for n = 2 ($D_4=V$ is abelian).