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

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