Unveiling Lagrange’s Theorem and its Applications in Proving other Theorems

Abstract

The group theory is an essential subject in Mathematics and Physics, and the target of this paper is to prove a famous theorem in group theory, so-called Lagrange’s Theorem. By using some really basic definitions of group, the exsistance of this theorem is essential for abstract algebra. In this paper, the author will focus on how to prove Lagrange’s Theorem step by step from the base of group theorey, mainly by using the nature of cosets and how does each coset in the same subgroup behaves to get the final result. Ultimately, the author will demonstrate that the order of the group is divisible by the order of its subset. As mentioned ealier, the status of this theorem is unshakable. Because of this theorem, many other corollary theorem was discovered, for example Wilson’s Theorem and etc. All of these corollaries are very important in modern technologys, going deep into this theorem could help discover more useful applications of it. This paper should be essential for people who are interested in the Lagrange’s theorem.