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

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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 $\gen {a^2, b}$ is normal.


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

First it is noted that:

\(\ds \paren {a^2}^2\) \(=\) \(\ds e\)
\(\ds b^2\) \(=\) \(\ds e\)


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

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.