Central Extensions and Groups with Quotients Periodic Infinite

For an arbitrary family of groups without involutions and any Abelian group D we construct a group AD(G) such that the center of AD(G) coincides with D, and the quotient group of the group AD(G) by the subgroup D coincides with the n-periodic product of the given family of groups. In particular, as an application, 2-generated non-simple and non-periodic Hopfian groups are constructed, any proper non-trivial quotient of which is infinite periodic. The construction is based on some modification of the method used by S.I. Adian for a positive solution of the known problem on the existence of non-commutative analogues of the additive group of rational numbers.