Lagrange’s Theorem in Group Theory

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