  Let p be a prime. Every finite group G has a normal series each of whose
quotients either is p-soluble or is a direct product of nonabelian simple
groups of orders divisible by p. The non-p-soluble length of G is defined as
the minimal number of non-p-soluble quotients in a series of this kind.
  We deal with the question whether, for a given prime p and a given proper
group variety V, there is a bound for the non-p-soluble length of finite groups
whose Sylow p-subgroups belong to V. Let the word w be a multilinear
commutator. In the present paper we answer the question in the affirmative in
the case where p is odd and the variety is the one of groups in which the
w-values have orders dividing a fixed number.
