Torsion in Engel modules

Abstract

The original motive for studying Lie rings with Engel condition stemmed from Burnside's problem. The restricted Burnside hypothesis (that any group of p on q generators has finite lower central series [8]) could be established for prime p by showing that any Lie nil ring of a certain kind on q generators is actually nilpotent. It is not known whether this sufficient condition is also necessary; nevertheless, there are advantages in treating the problem within the context of Lie rings. Transition from the group problem to the Lie-ring problem is set up by the Magnus representation. This maps the q group generators into the q generators of the Lie ring, while the significance in the Lie ring of the group's having p appears in two known ways. One is that, since the Magnus representation takes products into sums, the Lie ring must have pf = 0 for every elementf. The other lies deeper, and requires for the Lie ring that if f and g are any elements of L then [f, gp-r ] = [... [f, g], g, . . . , g] = O (the bracket denotes Lie multiplication; there are p -1 g's on the right). Because Engel's theorem (for Lie algebras) begins with such an assumption, a condition like this is called an Engel condition of p (though some writers prefer exponent p -1 ). In actual computations it is not enough to know that each [f, gp-l] =0. The information needed is that all of what will here be called S,'s, the strongly homogeneous parts of the polynomials [f, gp-1], are also zero. Over a field, as in computations directly bearing on Burnside's problem, a standard Vandermonde determinant argument shows that indeed, if all [f, gP-l] = 0, then also all such S, = 0. When the scalars may not admit division this is no longer necessarily so. The important role of the Engel condition in such computations has led to creation of an independent area of study in groups and rings subjected to an Engel condition (see, e.g., [4; 6]). To study the condition by itself, apart from the other consequence (pf = 0) of assuming p in the Burnside group, one may begin with a