Irreducible representations of some finite groups and galois stability of integral representations

Representation theory is a branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. A representation makes an abstract algebraic object more concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication. In this research, representations of certain finite groups are presented. The aims of this research are to find the matrix representations of dihedral groups, irreducible representations of dihedral groups and to relate the representations obtained with their isomorphic point groups. For these purposes, some theorems presented by previous researches on the representations of groups have been used. The fact that isomorphic groups have the same properties has also been applied in this research. Another part of this research is to explore on the representations of finite groups over algebraic number fields and their orders under field extension. Thus, this research also aims to prove the existence of abelian Galois stable subgroups under certain restriction of field extensions. The concept of generalized permutation modules has been used to determine the structure of the groups and their realization fields. Matrix representations of dihedral groups of degree six and the irreducible representations of all point groups which are isomorphic to the dihedral groups have been constructed. The existence of abelian Galois stable subgroups under certain restriction of field extensions have also been proven in this research