10.16 (1986)

A class of groups is called a direct variety if it is closed under taking subgroups, factor-groups, and direct products (Yu. M. Gorchakov, Groups with Finite Classes of Conjugate Elements, Moscow, Nauka, 1978 (Russian)). It is obvious that the class of $FC$-groups is a direct variety. P. Hall (J. London Math. Soc., 34, no. 3 (1959), 289–304) showed that the class of finite groups and the class of abelian groups taken together do not generate the class of $FC$-groups as a direct variety, and it was shown in (L. A. Kurdachenko, Ukrain. Math. J., 39, no. 3 (1987), 255–259) that the direct variety of $FC$-groups is also not generated by the class of groups with finite derived subgroups. Is the direct variety of $FC$-groups generated by the class of groups with finite derived subgroups together with the class of $FC$-groups having quasicyclic derived subgroups?