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.
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 $7.15$