First the composition maps .

We need to show,

Now,

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.

Next,

Thus,

Thus,

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,