Recall that, given groups the set of group homomorphisms is denoted by If then a group homomorphism is called an endomorphism and, in this case, we write instead of

**Problem**. i) Let be finite abelian groups. Find

ii) Let be a finite abelian group. Find and show that divides

iii) Let be a finite abelian group of order Show that if and only if

**Solution**. i) By the fundamental theorem for finite abelian groups, we have

for some integers Then by the problems in this post and this post, we have

and so, by the problem is this post,

That completes the solution of i).

Now let be a finite abelin group. Then, by the fundamental theorem for finite abelian groups,

for some integers where for all We can now solve ii) and ii).

ii) By i), we have

and so

Thus, since whenever we have

Since it’s clear from that divides

iii) Using in ii), we have

if and only if i.e. and so

**Example 1**. Let be a prime number and suppose that is a group of order Find and

**Solution**. We know that a group of order is abelian. Thus, by the fundamental theorem for finite abelian groups, either or

So, by the second part of the above problem, if then hence

and if then hence

**Example 2**. Find all finite abelian groups for which

**Solution**. Again, by the fundamental theorem for finite abelian groups,

for some integers where for all So, using the second part of the above problem, we want to have

That obviously has no solution for and for it is equivalent to

So the set of groups that satisfy the condition is

**Exercise**. Find all finite abelian groups for which