Elements of linear representation of finite groups.

Abstract

This paper is a survey of elementary concepts of the theory of representation of finite groups; wherein abstract groups are realized as groups of linear transformations or matrices. Basic definitions and examples of the above are given, as well as a notion of an equivalence relation for representations. The regular representation is presented through the concept of a group algebra. Other properties, such as sub-representations and irreducible representations, lead to an important result about the reducibility of representations of subgroups known as Clifford's Theorem. The character of a representation is then defined as the trace of the linear transformation or matrix which represents each element of the group. The relationship between characters and representations is developed including: (1) orthogonality relations from Schur's Lemma; and (2) the fact that the number of irreducible representations of a group is equal to its number of conjugacy classes. Finally, the concepts of induced representations and characters are explored, culminating in the Frobenius reciprocity formula.