# 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:

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

### Quotient Group

Let $E := N, A := a N, B := b N, C := a b N$.

Thus the quotient group of $G$ by $N$ is:

$G / N = \set {E, A, B, C}$

whose Cayley table can be presented as:

$\begin{array}{c|cccc} & E & A & B & C \\ \hline E & E & A & B & C \\ A & A & E & C & B \\ B & B & C & E & A \\ C & C & B & A & E \\ \end{array}$

which is seen to be an example of the Klein $4$-group.

## 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 $N$:

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

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

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

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

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.

$\blacksquare$