automorph

An automorphism is an isomorphism from a mathematical object to itself. This concept is used in various branches of mathematics, including abstract algebra, geometry, and topology. An automorphism is a bijective homomorphism, meaning it is a structure-preserving map between a mathematical object and itself, and is both one-to-one and onto.

In group theory, an automorphism is an isomorphism from a group to itself. Automorphisms of a group can be thought of as symmetries of the group structure. Automorphisms of a graph can be described in terms of the adjacency matrix of the graph.

Automorphisms are important in geometry, particularly in the study of symmetries of geometric objects. For instance, an automorphism of a plane geometry can describe a rotation or reflection of the plane. In topology, automorphisms are used to study the symmetries of topological spaces, which can provide information about the topology of the space.

Automorphisms can be classified into different types based on their properties. For example, an inner automorphism is an automorphism that can be expressed as conjugation by an element of the group. Automorphisms can also be described in terms of properties such as being solvable, nilpotent, or Abelian.

The study of automorphisms has numerous applications in mathematics and computer science. In coding theory, automorphisms of a code are used to study its symmetries and properties. In computational geometry, automorphisms are used to study the symmetries of geometric objects and calculate their structural properties. Automorphisms are used to study the symmetries of algebraic structures, such as groups and rings.