# Prove that G is abelian iff...

• Nov 15th 2009, 07:01 AM
Magics6
Prove that G is abelian iff...
Prove that G is abelian if and only if the map $\displaystyle f: G \longrightarrow G$ given by $\displaystyle f(g)=g^2$ is a group homomorphism.

$\displaystyle f(g_{1} \cdot g_{2})=g_{1}g_{2}g_{1}g_{2}=g_{1}g_{1}g_{2}g_{2}=g _{1}^2g_{2}^2$

if G abelian then f is homomorphism.
Nevertheless i prove only one direction of the statement. Can anybody help me?
• Nov 15th 2009, 07:11 AM
tonio
Quote:

Originally Posted by Magics6
Prove that G is abelian if and only if the map $\displaystyle f: G \longrightarrow G$ given by $\displaystyle f(g)=g^2$ is a group homomorphism.

$\displaystyle f(g_{1} \cdot g_{2})=g_{1}g_{2}g_{1}g_{2}=g_{1}g_{1}g_{2}g_{2}=g _{1}^2g_{2}^2$

if G abelian then f is homomorphism.
Nevertheless i prove only one direction of the statement. Can anybody help me?

If f is a homom. then for any $\displaystyle a\,,\,b\in\,G\,,\,\,abab=aabb$ . Now cancel stuff here and show abelianess.

Tonio