Groups with peridic relations, embeddings and varieties

We plan to prove that for any countable abelian group D and for a given family of groups there exists a group A such that the center of A is D, and the quotient group A/D is isomorphic to the n-periodic product of the given family, as well as to show that any abelian group can be embedded as the center into a group A so that the quotient group A/D is isomorphic to a given n-torsion group, to obtain sufficient conditions for the constructed groups A, ensuring that they are torsion-free. We will study the automorphisms of the endomorphism semigroup of the free Burnside groups of period 3 of finite rank and the automorphisms of the free group of period 3 of infinite rank, we will find out whether they are tame, that is, are induced by some automorphism of the free group of infinite rank. We plan to obtain an embedding method of recursive groups into finitely presented groups, to construct embeddings of countable groups with special properties into 2-generated groups, to study explicit sequences of Higman operations, to construct embeddings of groups and varieties of groups using wreath products, to investigate the hyperidentities classification problem, to investigate Boole-de-Morgan bilattices and free algebras of varieties of Boole-de-Morgan bilattices.