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Let p be a prime. Every finite group G has a normal series each of whosequotients either is p-soluble or is a direct product of nonabelian simplegroups of orders divisible by p. The non-p-soluble length of G is defined asthe 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 propergroup variety V, there is a bound for the non-p-soluble length of finite groupswhose Sylow p-subgroups belong to V. Let the word w be a multilinearcommutator. In the present paper we answer the question in the affirmative inthe case where p is odd and the variety is the one of groups in which thew-values have orders dividing a fixed number.

Nonsoluble Length Of Finite Groups with Commutators of Small Order