elliptic curves have been one of the many objects of studying due to their rich structure. we will discuss the group law on elliptic curves and the staggering fact that this group is abelian and finitely generated. knowing the structure of finitely generated abelian groups, we ask what the torsion piece of the group can be. mazur’s theorem will give us our answer! in discussion we will encounter modular curves, rational points on a variety of objects and attempt to get a sense of arithmetic geometry. this talk should be accessible to anyone but knowledge of the classification of finitely generated abelian groups and a little complex analysis may be useful in some sections.