Quaternion Group/Cayley Table

Cayley Table for Quaternion Group

The Cayley table for the quaternion group given with the group presentation:

$Q = \Dic 2 = \gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$

can be presented as:

$\begin{array}{r|rrrrrrrr}

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

\hline e & e & a & a^2 & a^3 & b & a b & a^2 b & a^3 b \\ a & a & a^2 & a^3 & e & a b & a^2 b & a^3 b & b \\ a^2 & a^2 & a^3 & e & a & a^2 b & a^3 b & b & a b \\ a^3 & a^3 & e & a & a^2 & a^3 b & b & a b & a^2 b \\ b & b & a^3 b & a^2 b & a b & a^2 & a & e & a^3 \\ a b & a b & b & a^3 b & a^2 b & a^3 & a^2 & a & e \\ a^2 b & a^2 b & a b & b & a^3 b & e & a^3 & a^2 & a \\ a^3 b & a^3 b & a^2 b & a b & b & a & e & a^3 & a^2 \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}{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.

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Conjugacy, Normal Subgroups, and Quotient Groups: $\S 46 \iota$