Abstract
A matrix group is a group in which the elements are square matrices (of the same dimension), the multiplication law is matrix multiplication, and the group inverse of an element is the matrix inverse. Because the inverse of an element is the matrix inverse, and the inverse of a group element must exist, only non-singular matrices can be elements of matrix groups, and that the group identity is the identity matrix. Most applications of group theory to physical problems are applications of representation theory. One reason is that representation theory reduces the abstract properties of groups to numbers, with which a physicist feels more at home. Hence, representation theory is, for a physicist, the most important aspect of group theory. This theory was developed at the turn of the century almost single-handedly by the German mathematician F. G. Frobenius. The concept of representation rests on the two concepts, matrix group and homomorphism. This chapter discusses finite matrix groups.
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