Infinite group theory

Infinite group theory studies mathematical structures called groups that have infinitely many elements, meaning they are unending in size. A group consists of a set of elements and an operation (like addition or multiplication) satisfying properties such as closure, associativity, identity, and invertibility. Infinite groups can model symmetries and transformations in complex systems, from geometric patterns to algebraic objects. They help mathematicians understand how these infinitely large collections behave, how they can be classified, and their applications across science, physics, and cryptography, providing insights into structures that extend beyond finite limitations.