We need to show,
because is a homomorphism, and thus, because is also a homomorphism. Q.E.D.
There is a problem with this question.2. If G is an abelian group, prove that the mapping θ: G > G' defined by θ(x)=x^-1 is an automorphism.
Because an automorphism is defined between the same groups but you did not mention that. Thus, I assume the groups are the same.
We need to show that,
is an isomorpism.
Simple, first this is a function between these two sets.
This show the map is one-to-one.
Next, for any we can choose (for it has an inverse) then, . Thus this map is onto.
Finally we show the homomorphism part.
In general the inverse for any group is,
But this group is abelian thus,