Automorphism: the self mapping isomorphic functions. 



Automorphism as I wrote in my last post is a type of isomorphism, or we can simply say, it is an extension of an isomorphic function.
To understand how deep automorphism is, you can imagine it to be a composite function where isomorphism is contained in homomorphism and automorphism is contained in isomorphism, therefore we can say automorphism is equally contained in homomorphism.
To define automorphism, we say, it is an isomorphism from an object to 'itself'.
Or we can say, an automorphism is a bijective homomorphism of an object to itself.
The identity morphism is called a trivial automorphism while a non-identity morphism is called a nontrivial automorphism.
Definitions of automorphism depends on the algebraic structure in question, for instance, in category theory, an automorphism is an endomorphism.
If $X$ is an object, then the automorphism of the object $X$ form a group under composition of morphisms, this group is called an automorphism group of $X$ that satisfies the following properties:
  1. Closure property
  2. Associative property
  3. Identity property and 
  4. Inverse property.
Some examples of automorphism. 
  1.  In set theory, the arbitrary permutation of elements in a set is an automorphism. 
  2. In elementary arithmetic, the set of integers $\mathbb{Z}$ is considered as a group under addition, and it is a nontrivial automorphism. It is also considered as a ring, in rings, it is has only the trivial automorphism. 
  3.  In group theory, we say a group automorphism is an isomorphism of a group to itself.
  4.  In linear algbera, we say an automorphism is an invertible linear operator on a vector space $V$.
  5.  In rings, a field automorphism is a bijective ring homomorphism from a field to itself.
  6. In graph theory, an automorphism of a graph is a permutation of the nodes that preserves the edges and non-edges.
  7. In metric geometry, an automorphism is a self isometry, remember an isometry is an isomorphism in geometry. The automorphism group is called the isometry group.
  8. In category of Riemann surfaces, an automorphism is a conformal map from a surface to itself.
  9. An automorphism of a differentiable manifold $M$ is a diffeomorphism from $M$ to itself. 
  10. in topology, an automorphism of a topological space is a homeomorphism of the space to itself, or self-homeomorphism.
NB: In topology, it is not sufficient for a morphism to be bijective to be an isomorphism.