Automorphisms

An automorphism of a mathematical object X is a bijection from X to itself that preserves the structure that defines X. In other words, it is a map that respects the operations and relations of X and carries X onto itself without changing its essential properties.

The set of all automorphisms of X, equipped with composition of maps, forms a group called the automorphism group of X, denoted Aut(X). The identity map serves as the identity element, and the inverse of any automorphism is again an automorphism.

In group theory, an automorphism is a bijective homomorphism f: G → G. Inner automorphisms are those given by conjugation, f_g(x) = g x g^{-1} for some g in G. The quotient Aut(G)/Inn(G) is the outer automorphism group, denoted Out(G).

For a finite cyclic group C_n, the automorphism group Aut(C_n) is isomorphic to the unit group (Z/nZ)×. More generally, automorphisms of a structure depend on its defining operations and relations, and they reveal the symmetries of that structure.

In field theory, Aut(F) is the group of field automorphisms: maps F → F that preserve addition and multiplication and send 1 to 1. For finite fields F_{p^n}, Aut(F_{p^n}) is cyclic of order n and is generated by the Frobenius map x ↦ x^p.

In ring theory, automorphisms are bijective ring homomorphisms. For the ring of integers Z, the only automorphism is the identity. Polynomial rings and other rings can have more intricate automorphism groups.

In graph theory, a graph automorphism is a permutation of the vertices that preserves adjacencies. The automorphism group of the complete graph K_n is isomorphic to the symmetric group S_n. Graph automorphisms encode the symmetries of a graph or a geometric object.