Finite Group Z3

The unique group of Order 3. It is both Abelian and Cyclic. Examples include the Point Groups $C_3$ and $D_3$ and the integers under addition modulo 3. The elements $A_i$ of the group satisfy ${A_i}^3=1$ where 1 is the Identity Element. The Cycle Graph is shown above, and the Multiplication Table is given below.

$Z_3$	1	$A$	$B$
1	1	$A$	$B$
$A$	$A$	$B$	1
$B$	$B$	1	$A$

The Conjugacy Classes are $\{1\}$, $\{A\}$,

$$A^{-1}AA = A$$

$$B^{-1}AB = A,$$

and $\{B\}$,

$$A^{-1}BA = B$$

$$B^{-1}BB = B.$$

The irreducible representation (Character Table) is therefore

$\Gamma$	1	$A$	$B$
$\Gamma_1$	1	1	1
$\Gamma_2$	1	1	$-1$
$\Gamma_3$	1	$-1$	1