Automorfism

Automorfism is a term used in mathematics, particularly in abstract algebra, to describe a special type of function called an endomorphism that is also an isomorphism. An endomorphism is a structure-preserving map from a mathematical object to itself. An isomorphism is a map that preserves the structure of the object and has an inverse map that also preserves the structure, meaning the object and its image are essentially the same. Therefore, an automorfism is a bijective endomorphism.

In simpler terms, an automorfism is a symmetry of a mathematical object. It's a way to rearrange the elements of the object without changing its fundamental structure or relationships between elements. For example, in geometry, rotations and reflections of a shape are often automorfisms. In group theory, an automorfism of a group is a bijective homomorphism from the group to itself. This means that if you perform the operation within the group, the result is the same whether you use the original elements or their images under the automorfism.

The set of all automorfisms of a given mathematical object forms a group under the operation of function composition. This group is known as the automorphism group of the object and is a fundamental concept for understanding the symmetries and structure of that object. Studying automorphism groups helps mathematicians classify and understand different mathematical structures.