Group Homomorphism

By homomorphism we mean a mapping from one algebraic system to 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 exist 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)

Comments

comments