Words, permutations, and the nonsolvable length of a finite group

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We study the impact of certain identities and probabilistic identities on the structure of finite groups. More specifically, let w be a nontrivial word in d distinct variables and let G be a finite group for which the word map w_G\colon G^d\rightarrow G has a fiber of size at least \rho|G|^d for some fixed \rho > 0 . We show that, for certain words w , this implies that G has a normal solvable subgroup of index bounded above in terms of w and \rho . We also show that, for a larger family of words w , this implies that the nonsolvable length of G is bounded above in terms of w and \rho , thus providing evidence in favor of a conjecture of Larsen. Along the way we obtain results of independent interest on permutation groups; e.g. we show, roughly, that most elements of large finite permutation groups have large support.