Parity Group is 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.

Proof
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}$

It is easily checked that the four group axioms hold for both instances.

As there is only one group of order 2, up to isomorphism, from Group of Prime Order Cyclic, we see that both examples of the parity group are isomorphic to $C_2$.