2 3 N ov 2 00 6 The problem of the classification of the nilpotent class 2 torsion free groups up

In this paper we consider the problem of classification of the nilpotent class 2 finitely generated torsion free groups up to the geometric equivalence. By a very easy technique it is proved that this problem is equivalent to the problem of classification of the complete (in the Maltsev sense) nilpotent torsion free finite rank groups up to the isomorphism. This result, allows us to once more comprehend the complication of the problem of the classification of the quasi-varieties of nilpotent class 2 groups. It is well known that the variety of a nilpotent class s (for every s ∈ N) groups is Noetherian. So the problem of the classification of the quasi-varieties generated even by a single nilpotent class 2 finitely generated torsion free group, is equivalent to the problem of classification of complete (in the Maltsev sense) nilpotent torsion free finite rank groups up to the isomorphism. 1 Historical review and methodology. The one of the question which naturaly appears in all algebraic studies, is the question of the classification (up to isomorphism) of the algebraic objects from some class. One of the classical example of this kind of results is the classification of the semisimple finite dimensional associative algebras over fields. Also we have the very easely observed classification of the finitely generated abelian groups. Both of these classification were achieved many years ago. About 60 years ago, the classification of the simple Lie algebras over C was achieved. The classification of the finite simple groups, is a newer result of this kind. This result requested a huge effort of many mathematicians.