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

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.

Proof

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

First it is noted that:

\(\ds \paren {a^2}^2\)

\(=\)

\(\ds e\)

\(\ds b^2\)

\(=\)

\(\ds e\)

so:

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

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.

$\blacksquare$

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $7$: Normal subgroups and quotient groups: Example $7.15$