Dihedral Group D4/Normal Subgroups/Subgroup Generated by a^2/Quotient Group
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Quotient Group of Normal Subgroup of the Dihedral Group $D_4$
Let the dihedral group $D_4$ be represented by its group presentation:
- $D_4 = \gen {a, b: a^4 = b^2 = e, a b = b a^{-1} }$
Consider the normal subgroup $\gen {a^2}$ of $D_4$:
- $\gen {a^2} = \set {e, a^2}$
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.
Subgroups of Quotient
The subgroups of $G / N$ are:
\(\ds E\) | \(=\) | \(\ds \set {e, a^2}\) | ||||||||||||
\(\ds \set {E, A}\) | \(=\) | \(\ds \set {e, a, a^2, a^3}\) | ||||||||||||
\(\ds \set {E, B}\) | \(=\) | \(\ds \set {e, b, a^2, b a^2}\) | ||||||||||||
\(\ds \set {E, C}\) | \(=\) | \(\ds \set {e, b a, a^2, b a^3}\) | ||||||||||||
\(\ds \set {E, A, B, C}\) | \(=\) | \(\ds D_4\) |
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $7$: Normal subgroups and quotient groups: Example $7.13$