Thanks a lot ! I think I got it. Let me write it and you can plz check.

I understand - "Note also that (i ==> (ii) ==> (iii)" In an abelian G every sub-group is normal.

Right, so here is my approach

1. Prime Factorization of d - Let

2. So we have sub-gorups

of order

respectively

3. Claim

is the desired sub-group of order = d

a. That it is a subgorup is obvious from your point (iii)

b. To prove

, I will use induction.

Basic logic of induction

Consider

is a sub-group of both

and

. But

, thus

=1

Hence the result by your point# (iv)

I think I have got it. If you could plz let me know if I missed/mistaked somewhere.

And, thanks for your help.