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

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 wG: Gd → G has a fiber of size at least ρ|G|d for some fixed ρ < 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 ρ. 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 ρ, 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.