Linear Methods in Nilpotent Groups

Abstract

The subject of this chapter is commutator calculation. It will be recalled that the commutator [x, y] of two elements x, y of a group is defined by the relation

$$ [x,y] = {{x}^{{ - 1}}}{{y}^{{ - 1}}}xy. $$

. We then have

$$ [xy,z] = {{[x,z]}^{y}}[y,z],\quad [x,yz] = [x,z]{{[x,y]}^{z}}. $$

. These relations are rather similar to the conditions for bilinearity of forms, and there are a number of ways of formalizing this similarity. Once this is done, commutator calculations can be done by linear methods. Several examples of theorems proved by this method will be given in this chapter.