Parity Group is Group
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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 $2$.
$\blacksquare$
Sources
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.3$: Example $11$
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: A Little Number Theory