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) |

© 1996-9

1999-05-26