For a quasi variety of algebras K, the Higman Theorem is said to be true if every recursively presented K-algebra is embeddable into a finitely presented K-algebra; the Generalized Higman Theorem is said to be true if any K-algebra which is recursively presented over its finitely generated subalgebra is embeddable into a K-algebra which is finitely presented over this subalgebra. We suggest certain general conditions on K under which the Higman Theorem implies the Generalized Higman Theorem; a finitely generated K-algebra (...) A is embeddable into every existentially closed K-algebra containing a finitely generated K-algebra B if and only if the word problem for A is Q-reducible to the word problem for B. The quasi varieties of groups, torsion-free groups, and semigroups satisfy these conditions. (shrink)

The main result is that for every recursively enumerable existential consistent theory Γ , there exists a finitely presented SQ-universal group H such that Γ is satisfied in every nontrivial quotient of H. Furthermore if Γ is satisfied in some group with a soluble word problem, then H can be taken with a soluble word problem. We characterize the finitely generated groups with soluble word problem as the finitely generated groups G for which there exists a finitely presented group H (...) all of the nontrivial quotients of which embed G. We prove also that for every countable group G, there exists a 2-finitely generated SQ-universal group H such that every nontrivial quotient of H embeds G. (shrink)