PROBLEMS OF CLASSIFYING ASSOCIATIVE OR LIE ALGEBRAS OVER A FIELD OF CHARACTERISTIC NOT TWO AND FINITE METABELIAN GROUPS ARE WILD ∗

Abstract. 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 p 3 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. Key words. groups