## Finite Group Z4

One of the two groups of Order 4. Like , it is Abelian, but unlike , it is a Cyclic. Examples include the Point Groups and and the Modulo Multiplication Groups and . Elements of the group satisfy , where 1 is the Identity Element, and two of the elements satisfy .

The Cycle Graph is shown above. The Multiplication Table for this group may be written in three equivalent ways--denoted here by , , and --by permuting the symbols used for the group elements.

 1 1 1 1 1 1

The Multiplication Table for is obtained from by interchanging and .

 1 1 1 1 1 1

The Multiplication Table for is obtained from by interchanging and .

 1 1 1 1 1 1

The Conjugacy Classes of are , ,

 (1) (2) (3)

,
 (4) (5) (6)

and .

The group may be given a reducible representation using Complex Numbers

 (7) (8) (9) (10)

or Real Matrices
 (11) (12) (13) (14)