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

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Quotient Group of Normal Subgroup of the Dihedral Group $D_4$

Let the dihedral group $D_4$ be represented by its group presentation:

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

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.

Subgroups of Quotient

The subgroups of $G / N$ are:

\(\ds E\) \(=\) \(\ds \set {e, a^2}\)
\(\ds \set {E, A}\) \(=\) \(\ds \set {e, a, a^2, a^3}\)
\(\ds \set {E, B}\) \(=\) \(\ds \set {e, b, a^2, b a^2}\)
\(\ds \set {E, C}\) \(=\) \(\ds \set {e, b a, a^2, b a^3}\)
\(\ds \set {E, A, B, C}\) \(=\) \(\ds D_4\)