Abstract
The aim of this thesis is to study some of the symmetries of a projective plane curve by describing the fixed points of the subgroups of the alternating group A_5. The group A_5, is the rotational symmetry group of the regular icosahedron or regular dodecahedron. In chapter 2 , we cover some basic facts about the alternating group A_5. This includes the conjugacy classes and the subgroups structure. In chapter 3, we give a representation of the isomorphic group, the projective special linear group of degree two, over the field of five elements. In chapter 4, we present the group A_5 over the complex field. In chapter 5, we calculate eigenvalues and eigenvectors of the elements of A_{5} over complex field. In chapter 6 we examine the stabilizer subgroups and the group orbits, and in chapter 7 we find a projective plane curve whose automorphism group is A_{5}.