Isomorphism: a special case of homomorphism. 

An isomorphic relation between two sets G and H showing the property of injection and inverse.


Isomorphism is a mapping between two sets of the same algebraic structure that can be reversed by an inverse mapping. In a nutshell, we can define an isomorphism as an homomorphism reversed by an inverse mapping or we can also say that an homomorphism is isomorphic 'if and only if' it is bijective(invertible function). 
The name isomorphism was coined from two Greek words "isos" meaning "equal" and "morphe" meaning "shape", therefore we say two mathematical structures are isomorphic if there exists an isomorphism between them and since isomorphism deals with equality, then two isomorphic structures must have the same properties. Isomorphism also have extensions, namely, 
  •  Automorphism, which is an isomorphism from an object to itself. 
  • Canonical isomorphism is a case where there is only one isomorphism between two structures or obejcts. 
Isomorphism are called different names in different areas of mathematics, for example :
  1.  In topology, it's called an homeomorphism, which is defined as an isomorphism of  topological spaces. 
  2. In linear transformation, isomorphism are defined between vector spaces and are specifed by inverse matrices. 
  3. In metric space, it's called an isometry, and defined as an isomorphism of metric spaces. 
  4.  In differential manifolds, isomorphism is called a diffeomorphism, and is defined as an isomorphism of spaces that are equiped with differential structure. 
  5. In abstract algebra, a permutation is defined as an automorphism of a set. 
  6.  while in geometry, isomorphism is called a transformation e.g when you hear rigid transformation, affine transformation or projective transformation in geometry, you should know they are referring to isomorphism of each structure. 
An example of isomorphism is:
Let $\mathbb{R}^+$ be a multiplicative group of positive real numbers and let $\mathbb{R}$ be the additive group of real numbers. A map between $\mathbb{R}^+$ and $\mathbb{R}$ is defined by the logarithm function $\log:\mathbb{R}^+\rightarrow\mathbb{R}$ satisfies $\log(ab)=\log{a}+\log{b}, \forall{a,b}\in\mathbb{R}^+$ this is known as a group homomorphism with inverse property, therefore it is an isomorphism of groups. 
Another example is the exponential function defined $\exp:\mathbb{R}\rightarrow\mathbb{R}^+$ satisfies $\exp(a+b)=(\exp{a})(\exp{b})$ for all $a,b\in\mathbb{R}$, this is also an homomorphism with inverse property. 
Now lets look at the $\log\exp$ identities, if $\log\exp{a}=a$ and $\exp\log{b}=b$, then it is clear that there exists a bijection in this definition and therefore, the identity is isomorphic.