Dihedral Group D4/Normal Subgroups/Subgroup Generated by a
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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 $\gen a$ is normal.
Proof
Let $N = \gen a$
First it is noted that as $a^4 = e$:
- $N = \set {e, a, a^2, a^3}$
and is cyclic.
The left cosets of $N$:
\(\ds e N\) | \(=\) | \(\ds e \set {e, a, a^2, a^3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {e, a, a^2, a^3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds N\) |
\(\ds b N\) | \(=\) | \(\ds b \set {e, a, a^2, a^3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {b, b a, b a^2, b a^3}\) |
As $\order {\gen a} = \order {D_4} / 2$ it follows from Subgroup of Index 2 is Normal that $\gen a$ is normal.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $7$: Normal subgroups and quotient groups: Example $7.13$