One of the two groups of Order 4. The name of this group derives from the fact that it is a Direct Product of two Subgroups. Like the group , is an Abelian Group. Unlike , however, it is not Cyclic. In addition to satisfying for each element , it also satisfies , where 1 is the Identity Element. Examples of the group include the Viergruppe, Point Groups , , and , and the Modulo Multiplication Groups and . That , the Residue Classes prime to 8 given by , are a group of type can be shown by verifying that

(1) |

(2) |

The Cycle Graph is shown above, and the multiplication table for the group is given below.

1 | ||||

1 | 1 | |||

1 | ||||

1 | ||||

1 |

The Conjugacy Classes are , ,

(3) | |||

(4) | |||

(5) |

,

(6) | |||

(7) |

and .

Now explicitly consider the elements of the Point Group.

In terms of the Viergruppe elements

A reducible representation using 2-D Real Matrices is

(8) | |||

(9) | |||

(10) | |||

(11) |

Another reducible representation using 3-D Real Matrices can be obtained from the symmetry elements of the group (1, , , and ) or group (1, , , and ). Place the axis along the -axis, in the - plane, and in the - plane.

(12) | |||

(13) | |||

(14) | |||

(15) |

In order to find the irreducible representations, note that the traces are given by and Therefore, there are at least three distinct Conjugacy Classes. However, we see from the Multiplication Table that there are actually four Conjugacy Classes, so group rule 5 requires that there must be four irreducible representations. By rule 1, we are looking for Positive Integers which satisfy

(16) |

(17) |

1 | ||||

1 | 1 | 1 | 1 | |

1 | 1 | |||

1 | 1 | |||

1 | 1 |

These can be put into a more familiar form by switching and , giving the Character Table

1 | ||||

1 | 1 | |||

1 | 1 | |||

1 | 1 | 1 | 1 | |

1 | 1 |

The matrices corresponding to this representation are now

(18) | |||

(19) | |||

(20) | |||

(21) |

which consist of the previous representation with an additional component. These matrices are now orthogonal, and the order equals the matrix dimension. As before, .

© 1996-9

1999-05-26