# Parity Group is Group

## Theorem

The parity group is in fact a group.

## Proof

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

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

From Prime Group is Cyclic, there is only one group of order $2$, up to isomorphism.

Thus both instantiations of the parity group are isomorphic to $C_2$, the cyclic group of order $2$.

$\blacksquare$