This thesis is an extension (and generalization) of works done by Connes and Consani. In recent decades mathematicans have discussed about group scheme in arithmatic geometry. A group scheme can be considered as a group in the category of schemes. In this dissertation, we consider affine group scheme over k, where k is a filed. There are three main ways viewing affine group schemes over k: 1) as representable functors from the category of k-algebras to grou 2) as commutative Hopf algebras over k; 3) as groups in the category of schemes over k.