In this term paper we will write about algebraic structure group. In Introduction we will cite the denition of a group, it's elementary characteristics and terms as Abelian group, subgroup, homomorsm and isomorsm of group. Furthermore, in second part we'll describe few dierent type of groups starting with normal and quotient/factor groups within which we will state and prove Lagrange's theorem and rst theorem of isomorsm. Next we will describe cyclic and at the end p-groups with which we associate the term of Sylow-s p-groups, Cauchy's theorem and three Sylow's theorems.