Dihedral Group D4/Cayley Table

Cayley Table for Dihedral Group $D_4$

The Cayley table for the dihedral group $D_4$, whose group presentation is:

$D_4 = \gen {a, b: a^4 = b^2 = e, a b = b a^{-1} }$

can be presented as:

$\begin{array}{l|cccccccc} & e & a & a^2 & a^3 & b & b a & b a^2 & b a^3 \\ \hline e & e & a & a^2 & a^3 & b & b a & b a^2 & b a^3 \\ a & a & a^2 & a^3 & e & b a^3 & b & b a & b a^2 \\ a^2 & a^2 & a^3 & e & a & b a^2 & b a^3 & b & b a \\ a^3 & a^3 & e & a & a^2 & b a & b a^2 & b a^3 & b \\ b & b & b a & b a^2 & b a^3 & e & a & a^2 & a^3 \\ b a & b a & b a^2 & b a^3 & b & a^3 & e & a & a^2 \\ b a^2 & b a^2 & b a^3 & b & b a & a^2 & a^3 & e & a \\ b a^3 & b a^3 & b & b a & b a^2 & a & a^2 & a^3 & e \end{array}$

Coset Decomposition of $\set {e, a^2}$

Presenting the above Cayley table with respect to the coset decomposition of the normal subgroup $\gen{a^2}$ gives:

$\begin{array}{l|cc|cc|cc|cc} & e & a^2 & a & a^3 & b & b a^2 & b a & b a^3 \\ \hline e & e & a^2 & a & a^3 & b & b a^2 & b a & b a^3 \\ a^2 & a^2 & e & a^3 & a & b a^2 & b & b a^3 & b a \\ \hline a & a & a^3 & a^2 & e & b a^3 & b a & b & b a^2 \\ a^3 & a^3 & a & e & a^2 & b a & b a^3 & b a^2 & b \\ \hline b & b & b a^2 & b a & b a^3 & e & a^2 & a & a^3 \\ b a^2 & b a^2 & b & b a^3 & b a & a^2 & e & a^3 & a \\ \hline b a & b a & b a^3 & b a^2 & b & a^3 & a & e & a^2 \\ b a^3 & b a^3 & b a & b & b a^2 & a & a^3 & a^2 & e \end{array}$

which is seen to be an example of the Klein $4$-group.

Coset Decomposition of $\set {e, a, a^2, a^3}$

Presenting the above Cayley table with respect to the coset decomposition of the normal subgroup $\gen a$ gives:

$\begin{array}{l|cccc|cccc} & e & a & a^2 & a^3 & b & b a & b a^2 & b a^3 \\ \hline e & e & a & a^2 & a^3 & b & b a & b a^2 & b a^3 \\ a & a & a^2 & a^3 & e & b a^3 & b & b a & b a^2 \\ a^2 & a^2 & a^3 & e & a & b a^2 & b a^3 & b & b a \\ a^3 & a^3 & e & a & a^2 & b a & b a^2 & b a^3 & b \\ \hline b & b & b a & b a^2 & b a^3 & e & a & a^2 & a^3 \\ b a & b a & b a^2 & b a^3 & b & a^3 & e & a & a^2 \\ b a^2 & b a^2 & b a^3 & b & b a & a^2 & a^3 & e & a \\ b a^3 & b a^3 & b & b a & b a^2 & a & a^2 & a^3 & e \end{array}$

Sources

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