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

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

Consider the normal subgroup $\gen {a^2}$ of $D_4$:


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

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.