Homomorphism of Groups: properties and examples.



In abstract mathematics (algebra), homomorphism can be defined as a map or relation between two algebraic structures of the same type e.g groups, rings or linear spaces. So today, we will be looking at homomorphism in terms of groups.

What is a group homomorphism?

Let $(G,*)$ and $(H,\circ)$ be groups, then a function $f:G\rightarrow{H}$ is a homomorphism if 
$f(x*y)=f(x)\circ{f(y)},\forall{x,y}\in{G}$. 
It should be noted that the group operation on the left hand side is that of $G$ while the group operation on the right hand side is that of $H$. The function $f$ also maps the identity element of $G$ to the identity element of $H$, $f(e_G)=e_H$, the same thing also applies to the inverses of $G$ and $H$, $f(x^{-1})=f(x)^{-1}$.

NOTE THE FOLLOWING TERMINOLOGIES 
  1. If $f$ is a one-to-one function, then $f$ is a Monomorphism. 
  2. If $f$ is a subjective function(onto), then $f$ is an Epimorphism. 
  3. A monomorphism from $(G,*)$ onto $(H,\circ)$ is an Isomorphism.
  4. Let $f:G\rightarrow{G}$ be a homomorphism, it is clear that the domain and codomain are the same, therefore, $f$ is an Endomorphism.
  5. If the endomorphism is bijective, then it is an Automorphism.
Examples of group homomorphism.

Example 1: Let $(G,*)$ be an arbitrary group and $H=\{e\}$, then the function $f:G\rightarrow{H}$ such that $f(x)=e$ for any $x\in{G}$ is a homomorphism.

Example 2: Consider $\mathbb{R}$, a set of real numbers under addition and $\mathbb{C}$, the set of complex numbers under multiplication with $|Z|=1$. Let $\Phi:\mathbb{R}\rightarrow\mathbb{C}$ be the map $\phi(x)=e^{i2\pi{x}}$. Then $\phi$ is a homomorphism.

Example 3: The exponential map yields a group homomorphism from the group of real numbers $R$ with addition to the group of non-zero real numbers $R^*$ with multiplication. The kernel is ${0}$ and the image consists of the positive real numbers.

Example 4: If $h:G→H$ and $k:H→K$ are group homomorphisms, then so is $k\circ{h}:G→K$. This shows that the class of all groups, together with group homomorphisms as morphisms, forms a category.