# 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

\[f\left( {x + y} \right) = 2\left( {x + y} \right) = 2x + 2y = f\left( x \right) + f\left( y \right)\]