##  [Automorphism](/automorphism) 

  ##  [Automorphism](https://mathlogic.quantumdictionary.io/automorphism-0) 

  

 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

An isomorphism from an object to itself: a bijective structure-preserving morphism a: X → X with an inverse a^{-1}, representing a symmetry or self-equivalence of the object.

 

 

 

 

 





 

 



 ##  [Automorphism](https://puremath.quantumdictionary.io/automorphism-1) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

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.

 

 

 

 

 





 

 



 ##  [Automorphism](https://algebra.quantumdictionary.io/automorphism-2) 

  

 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

An isomorphism from an algebraic object to itself: a bijective endomorphism whose inverse is also structure-preserving; represents a symmetry of the object.