The application of group and linear algebra theories to prove geometric transformation properties of the plane

The Geometric transformation set on R2 forms a group with function composition as the binary operation. Therefore, group theory can be used to derive properties applied to the set. This study focused on the Geometric transformation property on R2 based on the type of transformation, the existence of proper subgroup and normal subgroup, and the formation of factor group. Correspondingly, this research aims to derive properties that apply to elements or their subsets. Concepts and theories used to discuss and analyze the research problems were concepts and theorems in group theory, analytic geometry of the plane, and linear algebra. Hence, the scope of the set needs to be limited to examine the research problem using an algebraic approach. Furthermore, by applying concepts and theorems relevant to the research problems, we obtained the composition and inverse transformation formulas, the proof of the existence of proper subgroups and normal subgroups, and the proof of the factor groups isomorphic to GL (2, R).