Definition:Parity Group

Definition
This has several forms, all of which are isomorphic:

etc.
 * $C_2$, the cyclic group of order 2
 * The group $\left({\left\{{1, -1}\right\}, \times}\right)$
 * The group $\left({\Z_2, +_2}\right)$
 * The quotient group $\dfrac {S_n} {A_n}$ of the symmetric group of order $n$ with the altenating group of order $n$

It is the only group with two elements.

Cayley Table
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}$

Also see

 * Parity Group is Group