Finite Group Z4

One of the two groups of Order 4. Like $Z_2\otimes Z_2$, it is Abelian, but unlike $Z_2\otimes Z_2$, it is a Cyclic. Examples include the Point Groups $C_4$ and $S_4$ and the Modulo Multiplication Groups $M_5$ and $M_{10}$. Elements $A_i$ of the group satisfy ${A_i}^4=1$, where 1 is the Identity Element, and two of the elements satisfy ${A_i}^2=1$.

The Cycle Graph is shown above. The Multiplication Table for this group may be written in three equivalent ways--denoted here by $Z_4^{(1)}$, $Z_4^{(2)}$, and $Z_4^{(3)}$--by permuting the symbols used for the group elements.

$Z_4^{(1)}$	1	$A$	$B$	$C$
1	1	$A$	$B$	$C$
$A$	$A$	$B$	$C$	1
$B$	$B$	$C$	1	$A$
$C$	$C$	1	$A$	$B$

The Multiplication Table for $Z_4^{(2)}$ is obtained from $Z_4^{(1)}$ by interchanging $A$ and $B$.

${\mathcal Z}_4^{(2)}$	1	$A$	$B$	$C$
1	1	$A$	$B$	$C$
$A$	$A$	1	$C$	$B$
$B$	$B$	$C$	$A$	1
$C$	$C$	$B$	1	$A$

The Multiplication Table for $Z_4^{(3)}$ is obtained from $Z_4^{(1)}$ by interchanging $A$ and $C$.

$Z_4^{(3)}$	1	$A$	$B$	$C$
1	1	$A$	$B$	$C$
$A$	$A$	$C$	1	$B$
$B$	$B$	1	$C$	$A$
$C$	$C$	$B$	$A$	1

The Conjugacy Classes of $Z_4$ are $\{1\}$, $\{A\}$,

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

$$B^{-1}AB = A$$ (2)

$$C^{-1}AC = A,$$ (3)

$\{B\}$,

$$A^{-1}BA = B$$ (4)

$$B^{-1}BB = B$$ (5)

$$C^{-1}BC = B,$$ (6)

and $\{C\}$.

The group may be given a reducible representation using Complex Numbers

$$1 = 1$$ (7)

$$A = i$$ (8)

$$B = -1$$ (9)

$$C = -i,$$ (10)

or Real Matrices

$$1 = \left[\begin{array}{cc}1 & 0 \\ 0 & 1 \end{array}\right]$$ (11)

$$A = \left[\begin{array}{cc}0 & -1 \\ 1 & 0 \end{array}\right]$$ (12)

$$B = \left[\begin{array}{cc}-1 & 0 \\ 0 & -1 \end{array}\right]$$ (13)

$$C = \left[\begin{array}{cc}0 & 1 \\ -1 & 0 \end{array}\right].$$ (14)

See also Finite Group Z2Z2