Parity Group/Cayley Table

Cayley Table for Parity Group
We can completely describe the parity group by showing its Cayley table:


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

\left({\left\{{1, -1}\right\}, \times}\right) & 1 & -1\\ \hline 1 & 1 & -1 \\ -1 & -1 & 1 \\ \end{array} \qquad \begin{array}{r|rr} \left({\Z_2, +_2}\right) & \left[\!\left[{0}\right]\!\right]_2 & \left[\!\left[{1}\right]\!\right]_2\\ \hline \left[\!\left[{0}\right]\!\right]_2 & \left[\!\left[{0}\right]\!\right]_2 & \left[\!\left[{1}\right]\!\right]_2 \\ \left[\!\left[{1}\right]\!\right]_2 & \left[\!\left[{1}\right]\!\right]_2 & \left[\!\left[{0}\right]\!\right]_2 \\ \end{array} \qquad \begin{array}{r|rr} + & \text{even} & \text{odd} \\ \hline \text{even} & \text{even} & \text{odd} \\ \text{odd} & \text{odd} & \text{even} \\ \end{array}$