In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The group concept is central to abstract algebra: other well-known algebraic structures, such as rings, fields and vector spaces, can be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have undergone advances and have become thematic areas in their own right.
Various physical systems, such as crystals and the hydrogen atom, can be modeled by symmetry groups. Thus, group theory and the theory of closely related representation have many important applications in physics, chemistry, and the science of materials. Group theory is also central to public key cryptography.
One of the most important mathematical achievements of the twentieth century was the collaborative effort, which absorbed over 10,000 pages of journals and mostly published between 1960 and 1980, culminating in a complete classification of simple finite groups.