ABSTRACT
This chapter discusses the concept of a group, which is one of the most fundamental concepts in abstract algebra. Vector spaces are examples of algebraic objects that contain a linear structure, and they also always contain a group. The chapter provides definitions and examples of groups. Some groups possess a very rich “internal structure” and some groups have smaller groups called as subgroups. A subgroup must contain an identity element. The chapter presents a way to generate several subgroups within a given group and looks at the structure of the subgroups of a cyclic group. It defines cosets and studies some of their properties. The chapter proves a famous theorem of Lagrange which is the most important theorem in the study of finite groups. Lagrange’s theorem can be used to prove that subgroups of various orders cannot exist in a given group.
