# Dihedral Group D4/Normal Subgroups/Subgroup Generated by a^2/Quotient Group

Jump to navigation
Jump to search

## 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 $7.13$