We classify all complex 7- and 8-dimensional dual mock-Lie algebras by the algebraic and geometric way. Also, we find all non-trivial complex 9-dimensional dual mock-Lie algebras.

We show that p-groups of order p 5 are determined by their group algebras over the field of p elements. Many cases have been dealt with in earlier work of ourselves and others. The only case whose details remain to be given here is that of groups of nilpotency class 3 for p odd.

We investigate the group of normalized units of the group algebra Z p e G of a finite abelian p-group G over the ring Z p e of residues modulo p e with e ≥ 1.1991 Mathematics Subject Classification. Primary: 16S34, 16U60; Secondary: 20C05.

Let (A, B) be a pair of skew-symmetric matrices over a field of characteristic not 2. Its regularization decomposition is a direct sumis a pair of nonsingular matrices and (A 1 , B 1 ), . . . , (A t , B t ) are singular indecomposable canonical pairs of skew-symmetric matrices under congruence. We give an algorithm that constructs a regularization decomposition. We also give a constructive proof of the known canonical form of (A, B) under congruence over an algebraically closed field of characteristic not 2.

No abstract