automorphism

An automorphism of a mathematical structure is a bijection from the structure to itself that preserves all of its defining relations and operations. Aut(X) denotes the set (and usually the group under composition) of all automorphisms of X; its elements are the symmetries of X.

In group theory, an automorphism φ: G → G is a bijective homomorphism. The set Aut(G) forms a group under composition. Inner automorphisms, given by φ_g(x) = g x g^{-1}, form a normal subgroup Inn(G); the quotient Aut(G)/Inn(G) is the group of outer automorphisms.

In rings and fields, automorphisms preserve both addition and multiplication; field automorphisms also preserve the multiplicative identity. Notable examples include complex conjugation as a field automorphism of the complex numbers C that fixes the real numbers R, and, for finite fields F_q with q = p^n, the Frobenius automorphism x ↦ x^p whose powers generate all automorphisms. For vector spaces, automorphisms are invertible linear maps, forming the general linear group GL(V).

In graph theory, an automorphism of a graph permutes vertices in a way that preserves adjacency. The automorphism group of a graph encodes its symmetries; for example, the square (cycle C4) has dihedral symmetry D4.

Automorphisms generalize to many other structures, including modules, ordered sets, and topological spaces. They are central in studying symmetry, invariants, and classification, and their collection Aut(•) provides deep insight into the structure’s properties.