Let $G$ be a finite abelian group. I know of two ways of writing it as a direct sum of cyclic groups:

1) With orders $d_1, d_2, \ldots, d_k$ in such a way that $d_i|d_{i+1}$,

2) With orders that are powers of not necessarily distinct primes $p_1^{\alpha_1}, \ldots, p_n^{\alpha_n}$.

Is it true, and how can one prove that the cardinality $c$ of any minimal generating set for $G$ satisfies $k \leq c \leq n$ (I am most concerned about the second inequality)? Here minimal means irredundant.