Automorfizm

An automorphism is a structure-preserving isomorphism from a mathematical object to itself. In other words, it is a bijective (one-to-one and onto) mapping that establishes a correspondence between elements of a set or algebraic structure while preserving its defining properties. The concept of automorphisms is fundamental in abstract algebra, group theory, and other branches of mathematics.

In group theory, an automorphism of a group *G* is an isomorphism from *G* to itself. This means that an automorphism is a permutation of the elements of *G* that respects the group operation. For example, if *G* is a group with operation denoted by *, then an automorphism *φ* satisfies *φ(a * b) = φ(a) * φ(b)* for all *a, b* in *G*. The set of all automorphisms of *G*, denoted Aut(*G*), forms a group under composition of functions, known as the automorphism group of *G*.

Automorphisms also play a crucial role in other algebraic structures, such as rings, fields, and vector spaces. For instance, an automorphism of a field *F* is an isomorphism from *F* to itself that preserves addition and multiplication. Fields with non-trivial automorphisms are often studied in Galois theory, which explores the relationship between field extensions and group theory.

In graph theory, an automorphism is a permutation of the vertices of a graph that preserves adjacency. This means that if two vertices are connected by an edge, their images under the automorphism must also be connected by an edge. The set of all automorphisms of a graph forms a group called the automorphism group of the graph, which helps classify and study graph symmetries.

Automorphisms are also relevant in geometry, where they represent transformations that map a geometric object onto itself while preserving its properties. For example, in Euclidean geometry, rotations, reflections, and translations are types of automorphisms of the plane or space. The study of automorphisms helps uncover the inherent symmetries of mathematical structures, providing insights into their underlying structure and properties.