Order of the group

Here g is the order of the group (the number of symmetry operations in the group- 6 in this ease) and Xr(R) is the eharaeter for the partieular symmetry T whose eomponent in the direet produet is being ealeulated. 

For a given application of group theory it is furthermore necessary to both the set of elements X,Y,Z... and the law of combination ( multiplication or other). The following examples should clarify this question. It is to be noted that the order of the group is defined as the number of its members. 

Hie permutation of three identical objects was illustrated in Section However, in the application considered there, coordinate systems were used to specify the positions of the particles. It was therefore necessary on the basis of feasibility arguments to include the inversion of coordinates (specified by the symbol ) with those permutations that would otherwise change the handedness of the system. Nevertheless, for the permutation of three particles the order of the group was found to be equal to 3 = 6. 

This statement is often taken as a basic theorem of representation theory. It is found that for any symmetry group there is only one set of k integers (zero or positive), the sum of whose squares is equal to g, the order of the group. Hence, from Eq. (29), the number of times that each irreducible representation appears in the reduced representation, as well as its dimension, can be determined for any group. 

Computational effort for computing matrix elements with symmetry-projected basis functions can be reduced by a factor equal to the order of the group by exploiting commutation of the symmetry projectors with the Hamiltonian and identity operators. In general. 

The number of elements in a group is called the order of the group and is usually given the symbol g e.g. for the point group for the symmetric tripod g is 6 and for an infinite group g is oo. 

From these four rules it is easy, for example, to construct Table 7-7.1. There are three classes for jfSv and therefore three irreducible representations. The only three numbers whose squares add up to six (the order of the group) are 1,1, and 2. We therefore immediately have ... 

The proof that the sum of the squares of the dimensions of the irreducible representations is equal to the order of the group has three parts (1) intro, duction of the regular representation, (2) the Celebrated Theorem, (3) the final steps. 

The factor lj/h, where h is the order of the group and b 18 the dimension of the y th irreducible representation, has been included in (9.67) for convenience. Application of this procedure to the functions / gives us (unnormalized) symmetry-adapted functions g,. This procedure is applicable to generating sets of functions that form bases for irreducible representations from any set of functions that form a basis for a reducible representation. The proof of the procedure (9.67) for one-dimensional representations is outlined in Problem 9.22 we omit its general proof.5 Symmetry-adapted functions produced by (9.67) that belong to the same irreducible representation are not, in general, orthogonal. 

The sum of the squares of the absolute values of the characters in any irreducible representation is equal to the order of the group. 

Our discussions so far have tacitly implied that the order of the group g is a finite number, but this is not a necessary requirement, and we shall in fact deal with a number of infinite groups, as well as finite groups. Of most immediate importance to us, however, are groups of transformations that leave certain objects invariant, such as spatial rotations and reflections of a solid or an array of points, or transformations of functions. Before concluding this recapitulation of abstract group theory, therefore, we shall discuss some important aspects of groups of transformations. 

The number of elements g in G is called the order of the group. Thus... 

---