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

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Example of Normal Subgroup of the Dihedral Group $D_4$

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

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


The subgroup of $D_4$ generated by $\gen {a^2}$ is normal.


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.

$\blacksquare$


Sources