# Definition:Parity Group

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## Contents

## 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:

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

## Also see

## 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 $11$