Dihedral Group D4/Normal Subgroups/Subgroup Generated by a

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 $\gen a$ is normal.

Proof
Let $N = \gen a$

First it is noted that as $a^4 = e$:


 * $N = \set {e, a, a^2, a^3}$

and is cyclic.

The left cosets of $N$:

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