Dihedral Group D4/Normal Subgroups/Subgroup Generated by a^2

Example of Normal Subgroup of the Dihedral Group $D_4$
Let the dihedral group $D_4$ be represented by its group presentation:

The subgroup of $D_4$ generated by $\gen {a^2}$ is normal.

Proof
Let $N = \gen {a^2}$

First it is noted that as $\paren {a^2}^2 = a^4 = e$:


 * $N = \set {e, a^2}$

The left cosets of $N$:

As $b a^2 = a^2 b$ and $b a^3 = a^2 b a$, it follows immediately that:


 * $b N = N b, a N = N a, b a N = N b a$

and so $\gen {a^2}$ is seen to be normal.