A group is defined as a finite or infinite set of Operands (called ``elements'') , , , ... that may be combined or ``multiplied'' via a Binary Operator to form well-defined products and which furthermore satisfy the following conditions:

- 1. Closure: If and are two elements in , then the product is also in .
- 2. Associativity: The defined multiplication is associative, i.e., for all , .
- 3. Identity: There is an Identity Element (a.k.a. , , or ) such that for every element .
- 4. Inverse: There must be an inverse or reciprocal of each element. Therefore, the set must contain an element such that for each element of .

The study of groups is known as Group Theory. If there are a finite number of elements, the group is called a Finite Group and the number of elements is called the Order of the group.

Since each element , , , ..., , and is a member of the group, group property 1 requires that
the product

(1) |

(2) |

(3) |

(4) |

(5) |

(6) |

An Irreducible Representation of a group is a representation for which there exists no Unitary Transformation
which will transform the representation Matrix into block diagonal form. The Irreducible Representation has some
remarkable properties. Let the Order of a Group be , and the dimension of the th
representation (the order of each constituent matrix) be (a Positive Integer). Let any operation be denoted
, and let the th row and th column of the matrix corresponding to a matrix in the th Irreducible
Representation be
. The following properties can be derived from the Group Orthogonality Theorem,

(7) |

- 1. The Dimensionality Theorem:

(8) - 2. The sum of the squares of the Characters in any Irreducible Representation
equals ,

(9) - 3. Orthogonality of different representations

(10) - 4. In a given representation, reducible or irreducible, the Characters of all Matrices belonging to operations in the same class are identical (but differ from those in other representations).
- 5. The number of Irreducible Representations of a Group is equal to the number of Conjugacy Classes in the Group. This number is the dimension of the Matrix (although some may have zero elements).
- 6. A one-dimensional representation with all 1s (totally symmetric) will always exist for any Group.
- 7. A 1-D representation for a Group with elements expressed as Matrices can be found by taking the Characters of the Matrices.
- 8. The number of Irreducible Representations present in a reducible
representation is given by

(11)

(12)

**References**

Arfken, G. *Mathematical Methods for Physicists, 3rd ed.* Orlando, FL: Academic Press, pp. 237-276, 1985.

Farmer, D. *Groups and Symmetry.* Providence, RI: Amer. Math. Soc., 1995.

Weisstein, E. W. ``Groups.'' Mathematica notebook Groups.m.

Weyl, H. *The Classical Groups: Their Invariants and Representations.* Princeton, NJ: Princeton University Press, 1997.

Wybourne, B. G. *Classical Groups for Physicists.* New York: Wiley, 1974.

© 1996-9

1999-05-25