Prove a result on the size of the minimal set that generates a finite abelian group I am asked to prove the following:
Let $G$ be a non-trivial, finite abelian group. Let $s$ be the smallest positive integer such that $G = \langle a_1,...,a_s\rangle$ for some $a_1,...,a_s \in G$. Show that $s$ is equal to the number $t$ in the relation:
$$G \cong \mathbb{Z}_{m_1} \oplus\cdots\oplus \mathbb{Z}_{m_t}$$ 
where $m_i \mid m_{i+1}$ for $i=1,..,t-1$. 
We know that such a list of integers ($m_1,...,m_t $) exists, and is unique by 
The Fundamental Theorem of finite abelian groups.
I tried approaching the problem as follows:
Let $\rho : \mathbb{Z}_{m_1} \oplus\cdots\oplus \mathbb{Z}_{m_t} \rightarrow G$ be an isomorphism, then clearly:
$$G= \langle\ \rho(1,0,...0)\ ,\ \rho(0,1,...0)\ , ... ,\ \rho(0,0,...1)\ \rangle$$
, which implies $s\le t$. However I could not figure out how to show that $t \le s$.
 A: A clean way to do it consists perhaps in noting that if $p$ is a prime dividing $m_{1}$, then $G$ has a quotient group $Q$ isomorphic to $\mathbb{Z}_{p}^{t} = \mathbb{Z}_{p} \oplus \dots \oplus \mathbb{Z}_{p}$. 
If in
$$G \cong \mathbb{Z}_{m_1} \oplus\cdots\oplus \mathbb{Z}_{m_t}$$ 
a generator of the $\mathbb{Z}_{m_{i}}$ summand is $a_{i}$, then 
$$
Q = G / \langle p a_{1}, \dots , p a_{k} \rangle.
$$
In fact
$$
Q = \langle a_{1}, \dots , a_{k} \rangle / \langle p a_{1}, \dots , p a_{t} \rangle \cong \bigoplus_{i=1}^{t} \langle a_{i} \rangle / \langle p a_{i} \rangle \cong \mathbb{Z}_{p}^{t},
$$
as the order of each $a_{i}$ is a multiple of $p$.
Even without appealing to the theory of vector spaces (see the comment by OP below), a subgroup of $Q$ generated by $k$ elements is easily seen to have at most $p^{k}$ elements, so $Q$ cannot be generated by less than $t$ elements.
Thus $G$, too, cannot be generated by less than $t$ elements, as any set of generators for $G$ will induce a set of generators for $Q$.
Now, as OP already noted, the isomorphism $$G \cong \mathbb{Z}_{m_1} \oplus\cdots\oplus \mathbb{Z}_{m_t}$$  shows that $G$ has indeed a set of generators of size $t$.
