This diploma thesis is devoted to the groups and some theorems from group theory. The thesis is divided into two chapters. The first chapter consists of four sections. In the first section we define the group and some basic properties of groups (powers of group elements and group order). In the second section we define (proper) subgroups, cyclic subgroups, left and right cosets and index of a subgroup. An important theorem in this subsection is Lagrange’s theorem. In the third section we define a homomorphism, normal subgroups and the center of a group. In the fourth section we study normal subgroups and quotient groups. We state and prove three theorems on isomorphisms and define simple groups. The second chapter is titled as the work itself. At the beginning of this chapter we state and prove Zassenhaus or butterfly-lemma, we define normal and composition series, we define the equivalence of two normal series of the group and refinement of the normal sequence. We state and prove Schreier’s refinement theorem and as its consequence, we obtain the Jordan-Hölder theorem, which is one of the main results of thif thesis. We also define a comutator subgroups and central descending series of a group.