Lagrange’s Theorem in Group Theory

Abstract The structure and behavior of molecules crystals depend on their different symmetries. Thus, group theory is an essential technique in some fields chemistry. Within mathematics itself, very closely linked to symmetry geometry. Lagrange’s theorem a statement that can be viewed as extension the number theoretical result Euler’s theorem. It seen significant lemma for proving more complicated results theory. main intention this dissertation prove which illustrates every quadratic irrationality has periodic continued fraction. Conversely, fraction irrationality. first part paper research so-called Dirichlet groups, are subgroups preserving certain pairs lines. These groups related periodicity sails. induced by units order. Taking n-th roots two-dimensional matrices using Gauss’s reduction will also shown. Finally, solutions Pell’s equation proved.