Representations of Finite Groups

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The theory of group representation is concerned with group theory and linear algebra, with probably more of the latter than former. In Mathematics the word “representation” basically means “structure-preserving function”. Thus in group theory and ring theory at least a representation is a homomorphism. But more specifically, it should be a homomorphism from an object (group or ring) that one is trying to study to another that is in somewhat more concrete and hence, one hopes, easier to understand. The two simplest concrete kinds of groups are. firstly the group of all permutations of an arbitrary set, and secondly, the group of all invertible linear transformations on an arbitrary vector space. Groups were invented as a tool for studying symmetrical objects. These can be objects of any kind at all, if we define symmetry of an object to be a transformation of that object which preserves its essential structure then the set of all symmetries of the object forms a group. In mathematics it is always possible to regard any object as a set With Some additional structure-preserving objective function from the set to itself Composition of functions provides an operation on this set, and it is not hard to show that the group axioms must be satisfied. In order to understand groups one has to understand group actions. By considering function spaces these can be linearized, so it becomes important to understand linear actions on vector spaces, or representations of groups. Representation theory can be couched in terms of matrices or in the language of modules. We consider both approaches and then turn to the associated theory of characters.
A thesis submitted to the Board of Postgraduate Studies, Kwame Nkrumah University of Science and Technology in partial fulfilment of the requirements for the award of in Mathematics, 2004