GROUP

A group is a set equipped with a binary operation that combines any two elements to form a third, subject to four axioms: closure (the result stays in the set), associativity, existence of an identity element, and existence of inverses for every element. The operation is often denoted multiplicatively or additively.

In mathematics, groups are abstract structures used to study symmetry and other compositional processes. Examples include the integers under addition, the nonzero real numbers under multiplication, and the set of all permutations of a finite set under composition (the symmetric group). A group generated by a single element is called cyclic; a group is abelian if its operation is commutative. Finite groups have a finite number of elements, and their size is called the order. Subgroups are subsets that are themselves groups under the same operation; quotient groups arise from partitioning a group by a normal sub group. Homomorphisms are maps between groups that preserve the operation, while automorphisms are isomorphisms from a group to itself.

The term group is also used outside pure mathematics. In sociology and anthropology, a group is a collection of individuals who interact and share some identity, goal, or structure, such as a family, club, or work team. In geometry and physics, group concepts extend to Lie groups and topological groups, which carry additional structure that interacts with continuity and smoothness. The word therefore denotes both formal algebraic structures and broader collections defined by social or functional relations.