# Structure of finite abelian group

I am trying to solve this problem, and I think I should use the structure theorem for finite abelian groups, but i can't really figure this out.

Let $G$ be a finite abelian group. Prove there is a unique list $d_1, \ldots , d_n$ such that $d_1 > 1$ and $d_1|d_2 \ldots |d_n$, and $G \simeq \mathbb Z/ (d_1) \times \ldots \times \mathbb Z/(d_n)$;

Prove there is a list $p_1^{s_1}, \ldots p_t^{s^t}$ such that $p_1 \ldots p_t$ are primes, $s_1, \ldots s_t$ are positive, and $G \simeq \mathbb Z/p_1^{s_1} \ldots \mathbb Z/p_t^{s^t}$

Every pointer on how to solve this is appreciated, thanks again guys!

• How has the structure theorem been stated in your class? – Prahlad Vaidyanathan Oct 3 '13 at 4:12
• A key fact is that if $a$ and $b$ are relatively prime, then $\mathbb{Z}/(ab)$ is isomorphic to $\mathbb{Z}/(a)\times$\mathbb{Z}/(b)\$ (Chinese Remainder Theorem). – André Nicolas Oct 3 '13 at 4:25
• The structure theorem hasn't been stated, they gave us some questions saying we should search for the answers in algebraic geometry and algebraic number theory in order to stimulate our curiosity, maybe the structure theorem is not the right tool to use here? – user98379 Oct 3 '13 at 4:27
• The two statements in your question are the structure theorem for finite abelian groups. The first one is called invariant factor decomposition and the second one is called primary decomposition. – lhf Oct 3 '13 at 4:36
• THis seems much too hard for a problem to do on your own. Are you sure you're not missing something? – Alex Youcis Oct 3 '13 at 7:35