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 complex square roots of unity $\struct {\set {1, -1}, \times}$

The quotient group $\dfrac {S_n} {A_n}$ of the symmetric group of order $n$ with the alternating group of order $n$

and so on.

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}$

Also see

• Parity Group is Group
• Parity Group is Only Group with 2 Elements

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 7.4$. Kernel and image: Example $142$
• 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