Embedding arbitrary algebras in groups (Q1085190)

It is shown that there exists a group-theoretic word w in two variables and some constants (namely, the word \([a^ x,b^{x^{-1}}]\) in variables a, b and a constant x) which is ''universal'' in the sense that for any set A and binary operation \(f: A\times A\to A\), one can embed A in the underlying set of a group G, and choose values in G for the constants of w so that \(w(a,b)=f(a,b)\) for all a,b\(\in A\); and that here if A is finite, G can be taken finite as well. It is deduced that any finite family of finitary operations on a set A can be simultaneously represented by a family of group-theoretic words with constants, whose forms depend only on the system of arities of the operations. On the other hand, it is shown that such a representation of a binary operation is not always possible if G is required to be solvable. A similar construction is given satisfying the additional condition that the group structure on G extends any prescribed group structure on A. These constructions can be made functorial if one drops the restriction that G be finite when A is. Whether the functoriality and finiteness conditions can be obtained simultaneously remains an open question.