##  [Automorphism](/automorphism-1) 

 Definition

An isomorphism from a mathematical object to itself that preserves the object's specified algebraic or relational structure; it is a symmetry of that object.

 

 

 

 

 

 





## Principle

Principle

An automorphism is a bijective morphism f: X → X such that every operation or relation defining X is preserved; automorphisms compose to form a group under composition.

 

 

 

 

 





## Demonstration

Demonstration

For a finite group G, conjugation by a fixed g∈G defines an inner automorphism x↦gxg^{-1}; for a vector space V over a field, invertible linear maps form GL(V), the group of linear automorphisms.

 

 

 

 

## Misapplication

Misapplication

Calling any bijection of the underlying set an automorphism without checking preservation of operations (e.g., a random permutation of group elements generally fails to preserve the group law).

 

 

 

 

 





## Consequence

Consequence

Correctly identifying automorphisms yields the automorphism group Aut(X), which encodes symmetries, invariants, and classification data; fixed-point subobjects and orbit decompositions follow.

 

 

 

 

## Reversal

Reversal

A noninvertible self-homomorphism (endomorphism that is not an isomorphism) reverses the concept by allowing structure-preserving maps that are not bijections.

 

 

 

 

 





## Boundary

Boundary

Applies only to self-isomorphisms in the chosen category; excludes noninvertible endomorphisms and mappings between different objects; the notion depends on the ambient category and its notion of isomorphism.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension exists between ‘symmetry as bijection’ and ‘structure-preservation’; some authors relax bijectivity (e.g., approximate symmetries), which conflicts with the strict automorphism concept.

 

 

 

 

 





## Synthesis

Synthesis

An automorphism is the categorical symmetry of an object: a bijective structure-preserving self-map whose composition algebra forms the automorphism group and whose presence controls invariants and classification.