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In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity and invertibility. Many familiar mathematical structures such as number systems obey these axioms: for example, the integers endowed with the addition operation form a group. However, the abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way, while retaining their essential structural aspects. The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.^{[1]}^{[2]}
Groups share a fundamental kinship with the notion of symmetry. A symmetry group encodes symmetry features of a geometrical object: it consists of the set of transformations that leave the object unchanged, and the operation of combining two such transformations by performing one after the other. Such symmetry groups, particularly the continuous Lie groups, play an important role in many academic disciplines. Matrix groups, for example, can be used to understand fundamental physical laws underlying special relativity and symmetry phenomena in molecular chemistry.
The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—a very active mathematical discipline—studies groups in their own right.^{a[›]} To explore groups, mathematicians have devised various notions to break groups into smaller, betterunderstandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely (its group representations), both from a theoretical and a computational point of view. A particularly rich theory has been developed for finite groups, which culminated with the monumental classification of finite simple groups completed in 1983. Since the mid1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become a particularly active area in group theory.
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