If G is abelian and G rightarrow G is a homomorphism onto G

If G is abelian and: G rightarrow G\' is a homomorphism onto G\', prove that G\' is abelian.

Solution

Suppose a, b G¹. Then there are a¹, b¹ G such that (a¹) = a and (b¹) = b (because is surjective).

Thus ab = (a¹) (b¹) = (a¹ b¹) = (b¹ a¹) = (b¹) (a¹) = ba

Hence G¹ is abelian (we used both that is a homomorphism and that G is abelian).