  For a finite abelian group $G$, a nonempty subset $A$ of $G$, and a positive
integer $h$, we let $hA$ denote the $h$-fold sumset of $A$; that is, $hA$ is
the collection of sums of $h$ not-necessarily-distinct elements of $A$.
Furthermore, for a positive integer $s$, we set $[0,s] A=\cup_{h=0}^s h A$. We
say that $A$ is a generating set of $G$ if there is a positive integer $s$ for
which $[0,s] A=G$.
  The $h$-critical number $\chi (G,h)$ of $G$ is defined as the smallest
positive integer $m$ for which $hA=G$ holds for every $m$-subset $A$ of $G$;
similarly, $\chi (G,[0,s])$ is the smallest positive integer $m$ for which
$[0,s]A=G$ holds for every $m$-subset $A$ of $G$. We define $\widehat{\chi} (G,
h)$ as the smallest positive integer $m$ for which $hA=G$ holds for every
generating $m$-subset $A$ of $G$; $\widehat{\chi} (G, [0,s])$ is defined
similarly.
  The value of $\chi (G,h)$ has been determined by this author for all $G$ and
$h$, and $\widehat{\chi} (G, [0,s])$ was introduced and resolved for some
special cases by Klopsch and Lev. Here we determine the remaining two
quantities in all cases.
