# 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:

$D_4 = \gen {a, b: a^4 = b^2 = e, a b = b a^{-1} }$

## Proof

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

First it is noted that:

 $\ds \paren {a^2}^2$ $=$ $\ds e$ $\ds b^2$ $=$ $\ds e$

so:

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

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

The left cosets of $N$:

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

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

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.

$\blacksquare$