An upper bound for the nonsolvable length of a finite group in terms of its shortest law

TL;DR

This paper establishes an upper bound on the nonsolvable length of finite groups based on the length of their shortest law, linking algebraic properties with group structure and confirming a conjecture by Larsen.

Contribution

It proves a new theorem relating nonsolvable length to shortest law length, confirming Larsen's conjecture and addressing a problem by Khukhro and Shumyatsky.

Findings

Finite groups of nonsolvable length at least n do not satisfy non-trivial laws of length ≤ n.

A bound on nonsolvable length in terms of shortest law length is established.

The results confirm a conjecture of Larsen and provide a positive answer to a problem by Khukhro and Shumyatsky for p=2.

Abstract

Every finite group $G$ has a normal series each of whose factors is either a solvable group or a direct product of non-abelian simple groups. The minimum number of nonsolvable factors, attained on all possible such series in $G$, is called the \emph{nonsolvable length} $\lambda(G)$ of $G$. In the present paper, we prove a theorem about permutation representations of groups of fixed nonsolvable length. As a consequence, we show that in a finite group of nonsolvable length at least $n$, no non-trivial word of length at most $n$ (in any number of variables) can be a law. This result is then used to give a bound on $\lambda(G)$ in terms of the length of the shortest law of $G$, thus confirming a conjecture of Larsen. Moreover our Theorem C can be used to give a positive answer, in the case $p=2$, to a problem raised by Khukhro and Shumyatsky, concerning the non-$p$-solvable length of finite groups.