Automorfismi

Automorfismi, or automorphisms in English, are bijective maps from a mathematical object X to itself that preserve the structure defining X. In other words, an automorphism is an isomorphism from X to X. The collection Aut(X) of all automorphisms of X forms a group under composition, called the automorphism group of X, and this group encodes the internal symmetries of X.

In group theory, if X is a group G, an automorphism is a bijective homomorphism G → G. Inner automorphisms are those of the form φ_h(g) = hgh^{-1} for some h ∈ G; they form a normal subgroup Inn(G) of Aut(G). The quotient Aut(G)/Inn(G) is the outer automorphism group, describing symmetries not arising from conjugation.

In ring and field theory, a ring automorphism is a bijective ring homomorphism, and a field automorphism is a bijective field homomorphism. Field automorphisms typically fix the prime subfield. For finite fields GF(p^n), the automorphism group is cyclic, generated by the Frobenius map x ↦ x^p; indeed Aut(GF(p^n)) ≅ Z/nZ.

In graph theory, an automorphism is a permutation of the graph’s vertices that preserves adjacency. The automorphism group of a graph captures its symmetries, and similar notions apply in other contexts such as topology, where automorphisms are homeomorphisms, or in linear algebra for vector spaces and modules.

Examples include complex conjugation as a field automorphism of the complex numbers, and the automorphisms of a cyclic group of order n corresponding to units modulo n. Automorphisms are central to studying symmetry, invariants, and the structure of mathematical objects across disciplines.