I'll be glad to receive some help in the following question:
Let \(\displaystyle p \) be prime and let \(\displaystyle b_1 ,...,b_k \) be non-negative integers. Show that if :
\(\displaystyle G \simeq (\mathbb{Z} / p )^{b_1} \oplus ... \oplus (\mathbb{Z} / p^k ) ^ {b_k} \)
then the integers \(\displaystyle b_i \) are uniquely determined by G . (Hint: consider the kernel of the homomorphism \(\displaystyle f_i :G \to G \) that is multiplication by \(\displaystyle p^i \) . Show that \(\displaystyle f_1 , f_2 \) determine \(\displaystyle b_1 \). 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 !
Let \(\displaystyle p \) be prime and let \(\displaystyle b_1 ,...,b_k \) be non-negative integers. Show that if :
\(\displaystyle G \simeq (\mathbb{Z} / p )^{b_1} \oplus ... \oplus (\mathbb{Z} / p^k ) ^ {b_k} \)
then the integers \(\displaystyle b_i \) are uniquely determined by G . (Hint: consider the kernel of the homomorphism \(\displaystyle f_i :G \to G \) that is multiplication by \(\displaystyle p^i \) . Show that \(\displaystyle f_1 , f_2 \) determine \(\displaystyle b_1 \). 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 !