Dihedral Group D4/Cayley Table/Coset Decomposition of (e, a, a^2, a^3)

Cayley Table for Dihedral Group $D_4$

The Cayley table for the dihedral group $D_4$, with respect to the coset decomposition of the normal subgroup $\gen a$, is:

can be presented as:

$\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 $7.13$