Lagrange’s Theorem in Group Theory: Proof and Applications

Abstract

There are many propositions in group theory, among which Lagrange’s theorem is a representative example and its own meaning can be taken as a generalization of the Euler's theorem resulting from the number theory. Lagrange's theorem can be understood as follows. Suppose that  is a group and  denotes a subgroup of . This theorem clarify that the order of a subgroup  divides that of a group .The conjecture and discovery of Lagrange's theorem has led to the establishment of a unique framework in places such as calculus and statistics, allowing a certain interpretation for some problems. This paper presents the corollaries and applications of Lagrange's theorem, which can help to understand the properties of Lagrange's theorem easily, and can help to familiarize with how to apply Lagrange's theorem on various propositions. The basic concepts and fundamental theories of groups, rings and domains are the main contents of modern algebra, and the companion set and exponent are the most fundamental concepts in group theory.