groups

In mathematics, a group is a set G equipped with a binary operation that combines any two elements to form another element of G, satisfying four axioms: closure, associativity, the existence of an identity element, and the existence of inverses. Formally, a group (G, ·) satisfies: for all a, b, c in G, a·(b·c) = (a·b)·c; there exists e in G with e·a = a·e = a for all a; and for every a in G there is b in G with a·b = b·a = e. The operation is often written multiplicatively or additively.

Common examples include the integers under addition (Z, +), the nonzero rationals under multiplication (Q*, ×), and the symmetric group S_n, the group of all permutations of n elements under composition. A cyclic group is generated by a single element, such as Z_n under addition.

Subgroups and quotient groups: A subset H of G that is closed under the operation and contains the identity is a subgroup. If N is a normal subgroup of G, the quotient G/N carries a natural group structure. In finite groups, the order |G| is the number of elements, and Lagrange’s theorem states that the order of a subgroup divides |G|.

Morphisms: A group homomorphism f:G→H preserves the operation: f(a·b) = f(a)·f(b). Isomorphisms identify groups up to structure; automorphisms are isomorphisms from a group to itself.

Group actions: A group can act on a set, yielding orbits and stabilizers; the Orbit-Stabilizer theorem relates their sizes. Important classes include abelian groups, finite groups, p-groups, simple groups, and Lie groups.

Applications span mathematics, physics, chemistry, and cryptography. The term group also appears in the social sciences to denote a collection of individuals and their interactions.