Some Results of Character for Finite Groups

Abstract

The aim of our work is to study some of theories about the character.

The character theory is a very important theory within the concept of representation theory, especially in our study of finite groups, like symmetric group.

We touched on some basic definitions that we will need in our work, such as the definitions of homomorphism and isomorphism, the definition of character and the definition of class function, in addition to the definitions of tensor product, direct sum, and others.

We have proven that the character of direct sum is a class function. From this we came to the conclusion that the character of tensor product of the symmetric group is also a class function.

Then we proved that any number of direct sum of representations is an isomorphism if and only if they have the same character.

From the above theories, we were able to prove that the sum of three characters equals to the character of direct sum. And by the same way we also prove that the product of three characters equals to the character of tensor product. Representation theory is concerned with how to write a group as a collection of matrices.

The theory is not only beautiful in and of itself, but it also provides one of the keys to a proper understanding of finite groups. We have reached a critical point in the theory of finite group representation. The characters of a representation that we will study in this work will illustrate a great important about a representation. Also we have seen that a group can be represented by a set of invertible n×n matrices that show the same relationships as the group. Furthermore, we've seen that character and representation theories of finite

groups are extremely useful for studying and proving a variety of group theory

results. Obtaining the character table of a finite group is a difficult operation in general. This is deeply true when the degree of representation or the order for the finite group is very large, and the character only has one number for every matrix.

Character theory is the fundamental finding in group representation theory that establishes that linear representations of a finite group over the complex numbers disintegrate into irreducible parts. For instance, character table application relies on this essential concept. The theorem describes the excellent situations, for which the requisite behaviour will hold, in terms of the characteristics of the underlying field of scalars as an abstract algebraic construct. The real number field, for instance, has a similar pattern of behaviour; however the representations that are irreducible vary.