The number of generators of an abelian group of order $p^n$ with $p$ a prime

Question

My question is the following :

Let $G$ be an abelian group of order $p^n$, where $p$ is a prime and $n$ is a positive integer. Then by the fundamental theorem of finitely generated abelian groups, we can write $G=\Bbb Z_{p^{k_1}} \oplus$ ... $\oplus \Bbb Z_{p^{k_m}}$ where $k_1 \geq$ ... $\geq k_m \geq 1$ and $k_1 + $ ... $+k_m=n$. If $G$ is generated by $s$ elements, then must we have $s \geq m?$ My intuition says that this is true, but I don't know how to prove this.

Answer 1

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 we have $$A/pA\simeq(\mathbb Z/p\mathbb Z)^s,$$ so $|A/pA|=p^s$, and therefore $\log_p|A/pA|=s$. But this is the (minimal) number of generators.

This has been answered at this duplicate:

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