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
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $7$: Normal subgroups and quotient groups: Example $7.13$