Ela Problems of Classifying Associative or Lie Algebras over a Field of Characteristic Not Two and Finite Metabelian Groups Are Wild∗

Let F be a field of characteristic different from 2. It is shown that the problems of classifying (i) local commutative associative algebras over F with zero cube radical, (ii) Lie algebras over F with central commutator subalgebra of dimension 3, and (iii) finite p-groups of exponent p with central commutator subgroup of order p3 are hopeless since each of them contains • the problem of classifying symmetric bilinear mappings U × U → V , or • the problem of classifying skew-symmetric bilinear mappings U × U → V , in which U and V are vector spaces over F (consisting of p elements for p-groups (iii)) and V is 3-dimensional. The latter two problems are hopeless since they are wild; i.e., each of them contains the problem of classifying pairs of matrices over F up to similarity.