Alternating Trilinear Forms and Groups of Exponent

The theory of alternating bilinear forms on finite dimensional vector spaces V is well understood; two forms on V are equivalent if and only if they have equal ranks. The situation for alternating trilinear forms is much harder. This is partly because the number of forms of a given dimension is not independent of the underlying field and so there is no useful canonical description of an alternating trilinear form. In this paper we consider the set of all alternating trilinear forms on all finite dimensional vector spaces over a fixed finite field F and show that this set has a certain finiteness property. We then give a brief description of how this result may be used to prove two theorems on varieties of groups; in particular, that every group of exponent 6 has a finite basis for its laws. The details may be found in my D. Phil, thesis [1] which was written while I held a scholarship from the Science Research Council. This research was supervised by Dr. P. M. Neumann and Professor G. Higman for whose help I am heartily grateful.