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Representation theory for finite groups studies finite groups by embedding them in a general linear group over a field or an integral domain. Hence, we are representing each element of the group by an invertible matrix. Two marix representations of a given group are equivalent, if, by changing the basis of the underlying space, you can go from one to the other. When you change bases, you transform the matrices that are the images of elements by conjugating them by an invertible matrix.

If we can find a subspace which is fixed under the image of the group, then there exists a 'base change' after which all the representing matrices are in upper triangular block form. The block matrices on the main diagonal give a new representation of the group of lower degree. Such a representation is said to be 'reducible'.