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

## TL;DR

This paper investigates how certain identities influence the structure of finite groups, showing that specific word map properties imply bounds on the group's nonsolvable length, supporting a conjecture by Larsen.

## Contribution

It establishes new connections between word identities and the structural bounds of finite groups, particularly relating to nonsolvable length and solvable subgroups.

## Key findings

- Groups with large fiber size in word maps have bounded nonsolvable length.
- Certain words imply the existence of a normal solvable subgroup of bounded index.
- Most elements in large permutation groups have large support.

## Abstract

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 $w_G:G^d\rightarrow G$ 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 some independent interest, showing roughly that most elements of large finite permutation groups have large support.