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

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 $\gen {a^2, b}$ is normal.

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

First it is noted that:

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

and is an instance of the Kline $4$-group.

The left cosets of $N$:

As $\order {\gen {a^2, b} } = \order {D_4} / 2$ it follows from Subgroup of Index 2 is Normal that $\gen {a^2, b}$ is normal.