Proof and Applications of Lagranges Theorem in Deriving Fermats Little Theorem and Eulers Theorem

The group theory, as one of the cornerstones of the modern algebra, has a profound historical trajectory that reflects the evolution of the mathematical thought. This comprehensive paper analyses the historical development of the group theory and provides an overview of the interconnectedness of the several key theorems in the group theory: The Lagranges Theorem, the Fermats Little Theorem and the Eulers Theorem. This paper begins by establishing the modern group-theoretical framework within the Lagranges Theorem on the link between the order of groups and that of its subgroups. Then, an extension onto other related theorems are provided. In all, this paper is highly interlinking among the ideas in group theory. Ultimately, this study not only demonstrates the beauty of mathematical interconnections but also highlights their continuing relevance for the modern applications, showing how the classical results remain relevant to guide contemporary explorations in algebra, number theory, and related disciplines.