1.81 (1965)

The width of a group $G$ is, by definition, the smallest cardinal $m = m(G)$ with the property that any subgroup of $G$ generated by a finite set $S \subseteq G$ is generated by a subset of $S$ of cardinality at most $m$.
$\qquad$ a) Does a group of finite width satisfy the minimum condition for subgroups?
$\qquad$ b) Does a group with the minimum condition for subgroups have finite width?
$\qquad$ c) The same questions under the additional condition of local finiteness. In particular, is a locally finite group of finite width a Chernikov group?