automorfisms

Automorphy is a concept in mathematics that refers to the invariance of a mathematical structure or entity under a certain type of transformation. Specifically, an automorphism is a bijective homomorphism from a mathematical object to itself. In other words, it is an isomorphism from the object to itself.

Automorphisms are important in various areas of mathematics, including group theory, ring theory, and field theory. They can be used to describe symmetries of mathematical objects, such as geometric shapes or algebraic structures, that leave the object unchanged.

There are different types of automorphisms, including:

* Inner automorphisms: These are automorphisms that can be expressed as an inner operation, such as conjugation or inversion, of the mathematical object.

* Outer automorphisms: These are automorphisms that cannot be expressed as an inner operation and result in a change of the mathematical object's structure.

Automorphisms have numerous applications in mathematics and computer science, including:

* Cryptography: Automorphisms are used in cryptographic protocols, such as RSA, to ensure the secure transmission of data.

* Geometry: Automorphisms are used to describe symmetries of geometric shapes, such as crystals or molecules.

* Algebraic geometry: Automorphisms are used to study the properties and structure of algebraic varieties.

The study of automorphisms has been a topic of interest in mathematics for centuries, with contributions from numerous mathematicians, including Évariste Galois and Emmy Noether. Continued research in automorphisms has led to a deeper understanding of mathematical structures and has found applications in various fields.