The theory of groups, also known as group theory, is a branch of mathematics that studies algebraic structures called groups. A group consists of a set of elements combined with an operation that satisfies four key properties: closure, associativity, identity, and invertibility. This framework allows mathematicians to analyze symmetry, transformations, and other structures in various fields. Group theory has applications in many areas, including physics, chemistry, and computer science. It helps in understanding the symmetry of molecules, the behavior of particles, and even algorithms in cryptography. By studying groups, researchers can uncover deeper insights into the underlying patterns of complex systems.