# Definition:Parity Group

## Definition

This has several forms, all of which are isomorphic:

The group $\struct {\Z_2, +_2}$
$C_2$, the cyclic group of order 2
The group $\struct {\set {1, -1}, \times}$
The quotient group $\dfrac {S_n} {A_n}$ of the symmetric group of order $n$ with the alternating group of order $n$

etc.

It is the only group with two elements.

## Cayley Table

We can completely describe the parity group by showing its Cayley table:

$\begin{array} {r|rr} \struct {\set {1, -1} , \times} & 1 & -1\\ \hline 1 & 1 & -1 \\ -1 & -1 & 1 \\ \end{array} \qquad \begin{array} {r|rr} \struct {\Z_2, +_2} & \eqclass 0 2 & \eqclass 1 2 \\ \hline \eqclass 0 2 & \eqclass 0 2 & \eqclass 1 2 \\ \eqclass 1 2 & \eqclass 1 2 & \eqclass 0 2 \\ \end{array} \qquad \begin{array}{r|rr} + & \text{even} & \text{odd} \\ \hline \text{even} & \text{even} & \text{odd} \\ \text{odd} & \text{odd} & \text{even} \\ \end{array}$