# Abelian group homomorphism

• Nov 16th 2010, 04:16 PM
kathrynmath
Abelian group homomorphism
Let G be an abelian group and let n be any positive integer. Show that the function c:G--->G defined by c(x)=x^n is a homomorphism.

I started out by thinking of the definition of a homomorphism. We have a homomorphism if c(ab)=c(a)c(b).
So c(a)c(b)
a^nb^n
since G is abelian group, a^nb^n=(ab)^n=c(ab)?
• Nov 16th 2010, 05:26 PM
Drexel28
Quote:

Originally Posted by kathrynmath
Let G be an abelian group and let n be any positive integer. Show that the function c:G--->G defined by c(x)=x^n is a homomorphism.

I started out by thinking of the definition of a homomorphism. We have a homomorphism if c(ab)=c(a)c(b).
So c(a)c(b)
a^nb^n
since G is abelian group, a^nb^n=(ab)^n=c(ab)?

Right, I think. You can show by induction that since $\displaystyle G$ is abelian that $\displaystyle (ab)^{\text{anything}}=a^{\text{anything}}b^{\text {anything}}$ which is precisely what it means for $\displaystyle x^{\text{anything}}$ to be a homo.