Group of matrices

The ensemble of elements that are mutually conjugate form a class TJje concept of a class is most easily demonstrated by an example. The multiplication table for the group of matrices defined by Eq. (2) is given in Table 3. With its use the relations... 

Arbitrary groups of matrices are not our main concern, but we should record some simple relations between such groups and their closures. Apart from allowing more general statements of some later theorems, this will be useful because extension to a larger field involves taking-closures. 

For every finite group there is a group of matrices onto which the group elements are homomorphic (see Example 3.2). It is evident that a set of linear transformation matrices is isomorphic with the set of elements of any point symmetry group. For example, in Cs the following correspondences exist between the group operations and the 3X3 matrices which transform any column vector (x, y, z) according to the prescription of the operation ... 

These mappings of 3D space are the mappings represented by orthogonal matrices, i.e. they are elements of the group of matrices representing linear isometries, also denoted by O3, for sake of simplicity of notation ... 

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