In Charles C. Pinter's A Book of Abstract Algebra the following definition is given.
(D1) For every $x\in G$, there are integers $k_1, ...,k_n$ such that $x=a_1^{k_1}...a_n^{k_n}$
(D2) If there are integers $l_1,...,l_n$ such that if $a_1^{l_1}...a_n^{l_n}=e$ then $a_1^{l_1}=...=a_n^{l_n}$.
If (D1) and (D2) hold, we will write $G=[a_1,...,a_n]$.
Later on, Pinter writes the following.
Let $p$ be a prime number, and assume $G$ is a finite abelian group such that the order of every element in G is some power of $p$. Let $a\in G$ be an element whose order is the highest possible in $G$. Let $H=<a>$.
Explain why we may assume that $G/H=[Hb_1,...,Hb_n]$ for some $b_1,...,b_n \in G$.
So far I have deduced that (D1) is equivalent to saying that for every $x$ there exist integers $k_1,...,k_n$ such that $b_1^{k_1}...b_n^{k_n}x^{-1} \in H$. (D2) is the assertion that $b_1^{l_1}...b_n^{l_n} \in H \implies b_1^{l_1},...,b_n^{l_n} \in H$.
From here I really don't know where to go. It isn't obvious to me that such $b_1,...,b_n$ must exist, and I've tried to find elements that satisfy this property, but to no avail.