An algebraic structure (G, ·) consisting of a set G together with a binary operation · that is closed, associative, has an identity element e in G, and for every element g in G an inverse g^{-1} in G. The structure is considered up to equality of the set and operation, and homomorphisms preserve the operation.

Principle

A group organizes elements by a closed associative law with a neutral element and two-sided inverses; these axioms allow composition, cancellation, and the solution of equations within the set.

Demonstration

The integers Z with addition (Z, +) form a group: addition is associative and closed, 0 is the identity, and each n has inverse −n. Another standard example is the permutation group S_n of all bijections on an n-element set, with composition as the operation.

Misapplication

Treating any associative algebraic structure with an identity as a group while ignoring inverses (for example, calling a monoid a group) or assuming commutativity without proof are common misuses.

Consequence

When the group axioms hold, one obtains cancelation laws, unique identity and inverses, well-defined quotient groups by normal subgroups, and a rich theory of homomorphisms and group actions that connect algebra to geometry and combinatorics.

Reversal

Dropping inverses produces a monoid; dropping identity produces a semigroup; dropping associativity yields far weaker structures. Each omission weakens the algebraic control and invalidates many group-based arguments.

Boundary

This entry treats abstract groups only; it excludes extra structure such as topology, smooth structure, or algebraic-variety structure that leads to topological, Lie, or algebraic groups. Partial binary operations, multivalued operations, and structures lacking two-sided inverses fall outside this scope.

Semantic Tension

‘Group’ as an abstract four-axiom structure can be confused with specialized notions (topological group, algebraic group, group action). The tension is between pure axiomatic content and added geometric or analytic structure that imposes continuity, differentiability, or scheme-theoretic constraints.

Synthesis

A group is the basic closed, associative algebraic system with an identity and inverses; it serves as a minimal setting for symmetry, solvability of equations, and construction of quotients and homomorphisms.