Group theories are mathematical frameworks that study the algebraic structures known as groups. A group consists of a set of elements combined with an operation that satisfies four key properties: closure, associativity, identity, and invertibility. These properties allow for the analysis of symmetry and transformations in various mathematical contexts. In addition to pure mathematics, group theories have applications in fields like physics, chemistry, and computer science. They help explain phenomena such as particle interactions in quantum mechanics and molecular symmetries in chemistry. By understanding group structures, researchers can simplify complex problems and uncover underlying patterns.