On a problem from the Kourovka Notebook

In this article, it is proved that if a group $G$ coincides with its commutator subgroup, is generated by a finite set of classes of conjugate elements, and contains a proper minimal normal subgroup $A$ such that the factor group $G/A$ coincides with the normal closure of one element, then $G$ coincides with the normal closure of an element. From this a positive answer to question 5.52 from the Kourovka Notebook for the group with the condition of minimality on normal subgroups follows. We have found a necessary and sufficient condition for a group coinciding with its commutator subgroup and generated by a finite set of classes of conjugate elements not to coincide with the normal closure of any element.