Quaternion Group/Cayley Table/Coset Decomposition of (e, a^2)

Cayley Table for Quaternion Group $Q$

The Cayley table for the quaternion group $Q$, with respect to the coset decomposition of the normal subgroup $\gen a^2$, is:

can be presented as:

$\begin{array}{r|rr|rr|rr|rr}

     & e     & a^2   & a     & a^3   & b     & a^2 b & a b   & a^3 b \\

\hline e & e & a^2 & a & a^3 & b & a^2 b & a b & a^3 b \\ a^2 & a^2 & e & a^3 & a & a^2 b & b & a^3 b & a b \\ \hline a & a & a^3 & a^2 & e & a b & a^3 b & a^2 b & b \\ a^3 & a^3 & a & e & a^2 & a^3 b & a b & b & a^2 b \\ \hline b & b & a^2 b & a^3 b & a b & a^2 & e & a & a^3 \\ a^2 b & a^2 b & b & a b & a^3 b & e & a^2 & a^3 & a \\ \hline a b & a b & a^3 b & b & a^2 b & a^3 & a & a^2 & e \\ a^3 b & a^3 b & a b & a^2 b & b & a & a^3 & e & a^2 \end{array}$

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