In
mathematics, a
group is an
algebraic structure consisting of a
set of
elements equipped with an
operation that combines any two elements to form a third element. The operation satisfies four conditions called the group
axioms, namely
closure,
associativity,
identity and
invertibility. One of the most familiar examples of a group is the set of
integers together with the
addition operation, but the abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, applies much more widely. It 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.