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:
$\quad \begin {array} {r|rrrrrrrr} & e & a & a^2 & a^3 & b & a b & a^2 b & a^3 b \\ \hline e & e & a & a^2 & a^3 & b & a b & a^2 b & a^3 b \\ a & a & a^2 & a^3 & e & a b & a^2 b & a^3 b & b \\ a^2 & a^2 & a^3 & e & a & a^2 b & a^3 b & b & a b \\ a^3 & a^3 & e & a & a^2 & a^3 b & b & a b & a^2 b \\ b & b & a^3 b & a^2 b & a b & a^2 & a & e & a^3 \\ a b & a b & b & a^3 b & a^2 b & a^3 & a^2 & a & e \\ a^2 b & a^2 b & a b & b & a^3 b & e & a^3 & a^2 & a \\ a^3 b & a^3 b & a^2 b & a b & b & a & e & a^3 & a^2 \end {array}$
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:
$\quad \begin {array} {r|rr|rr|rr|rr}
& e & a^2 & a & a^3 & b & a^2 b & a b & a^3 b \\
\hline
e & e & a^2 & a & a^3 & b & a^2 b & a b & a^3 b \\
a^2 & a^2 & e & a^3 & a & a^2 b & b & a^3 b & a b \\
\hline
a & a & a^3 & a^2 & e & a b & a^3 b & a^2 b & b \\
a^3 & a^3 & a & e & a^2 & a^3 b & a b & b & a^2 b \\
\hline
b & b & a^2 b & a^3 b & a b & a^2 & e & a & a^3 \\
a^2 b & a^2 b & b & a b & a^3 b & e & a^2 & a^3 & a \\
\hline
a b & a b & a^3 b & b & a^2 b & a^3 & a & a^2 & e \\
a^3 b & a^3 b & a b & a^2 b & b & a & a^3 & e & a^2
\end {array}$
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: $\S 46 \iota$