I'll be glad to receive some help in the following question:

Let  p be prime and let  b_1 ,...,b_k be non-negative integers. Show that if :
 G \simeq (\mathbb{Z} / p )^{b_1} \oplus ... \oplus (\mathbb{Z} / p^k ) ^ {b_k}
then the integers  b_i are uniquely determined by G . (Hint: consider the kernel of the homomorphism  f_i :G \to G that is multiplication by  p^i . Show that  f_1 , f_2 determine  b_1 . Proceed similarly )

I've tried considering the mentioned homomorphisms, but without any success... I'll be delighted to receive some guidance/solution to this problem (that is some kind of a preliminary step towards the proof of the classification theorem of abelian groups).


Thanks in advance !