Group Homomorphism

By homomorphism we mean a mapping from one algebraic system with a like algebraic system which preserves structures.

Definition
Let $G$ and $G’$ be any two groups with binary operation $\circ$ and $\circ ‘$ respectively. Then a mapping $f:G \to G’$ is said to be a homomorphism if for all $a,b \in G$,
$f\left( {a \circ b} \right) = f\left( a \right) \circ ‘f\left( b \right)$

A homomorphism $f$ which at the same time is also onto is said to be an epimorphism.

A homomorphism $f$ which at the same time is also one-one is said to be an monomorphism.

A group $G’$ is called a homomorphism image of a group $G$, if there exists a homomorphism $f$ of $G$ onto $G’$. A homomorphism of a group $G$ into itself is called an edomorphism.

Examples:

(i) Let $G$ be any group under binary operation $\circ$. If $f\left( x \right) = x$ for every $x \in G$ then $f:G \to G$ is a homomorphism because $f\left( {xy} \right) = f\left( x \right)f\left( y \right)$

(ii) Let $G$ be the group of integers under addition, let $G’$ be the group of integers under addition modulo $n$. If $f:G \to G’$ be defined by $f\left( x \right) =$remainder of $x$ on division by $n$, then this is a homomorphism.

(iii) Let $G$ be any group under addition. If $f\left( x \right) = e$, $\forall x \in G$ then the mapping $f:G \to G$ is a homomorphism because for all $x,y \in G$, $f\left( {x,y} \right) = e$ and $f\left( x \right) + f\left( y \right) = e + e = e$, so that $f\left( {x + y} \right) = f\left( x \right) + f\left( y \right)$

(iv) Let $G$ be the group of integers under addition and let $G’ = G$. If for all $x \in G$, $f\left( x \right) = 2x$, then $f$ is a homomorphism because
$f\left( {x + y} \right) = 2\left( {x + y} \right) = 2x + 2y = f\left( x \right) + f\left( y \right)$