A reduction theorem for nonsolvable finite groups

TL;DR

This paper introduces the concept of n-rarefied groups and proves that every finite group with a certain nonsolvable length contains such a subgroup, leading to improved bounds on nonsolvable length in various group classes.

Contribution

It defines n-rarefied groups and proves their existence within finite groups of a given nonsolvable length, enhancing understanding of group structure.

Findings

Every finite group of nonsolvable length n contains an n-rarefied subgroup.

Improved upper bounds on the nonsolvable length of finite groups.

Determined maximum nonsolvable length for permutation and linear groups of fixed degree or dimension.

Abstract

Every finite group $G$ has a normal series each of whose factors is either a solvable group or a direct product of nonabelian simple groups. The minimum number of nonsolvable factors attained on all possible such series is called the nonsolvable length of the group and denoted by $\lambda(G)$. For every integer $n$, we define a particular class of groups of nonsolvable length $n$, called \emph{$n$-rarefied}, and we show that every finite group of nonsolvable length $n$ contains an $n$-rarefied subgroup. As applications of this result, we improve the known upper bounds on $\lambda(G)$ and determine the maximum possible nonsolvable length for permutation groups and linear groups of fixed degree resp. dimension.