# Minimal number of generators for a finitely generated abelian $p$-group

Let $A = \text{Tor}_p(A)$ be a finitely generated abelian $p$-group. (Here $p$ is prime). Show that the minimal number of generators of $A$ is $\log_p|A/pA|$.

What I tried -

I think that from the fundamental theorem of finitely generated abelian groups A is isomorphic to a group of the form $\mathbb Z_p\oplus\mathbb Z_p \oplus \cdots$, so it can be represented as an $n$-dimensional vector space.

I'm not sure if it is right and how to continue from here.

Any help will be appreciated.

• $\mathbb{Z}/(4)$ is also a $2$-group. – Loki Clock Dec 28 '15 at 11:01

## 1 Answer

A finitely generated abelian $p$-group $A$ looks like $$\mathbb Z/p^{k_1}\mathbb Z\oplus\cdots\oplus\mathbb Z/p^{k_s}\mathbb Z$$ with $k_1\ge\cdots\ge k_s\ge 1$. Then $$A/pA\simeq(\mathbb Z/p\mathbb Z)^s,$$ so $|A/pA|=p^s$, and therefore $\log_p|A/pA|=s$.

Now I leave you the pleasure to find out why $A$ can't have less than $s$ generators.

• still not sure about the generators.. Is it because A represent an s dimensional vector space, so it is not possible that it has less than s generators?? thanks. – user101010 Dec 29 '15 at 20:57
• @gpgpgp $A/pA$ is an $s$-dimensional vector space over $\mathbb Z/p\mathbb Z$. What's going on if $A$ has less than $s$ generators? – user26857 Dec 29 '15 at 21:07