A reduction theorem for nonsolvable finite groups

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 λ(G). For every integer n, we define a particular class of groups of nonsolvable length n, called 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 λ(G) and determine the maximum possible nonsolvable length for permutation groups and linear groups of fixed degree resp. dimension.

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

“…Clearly, a better bound might hold. In fact, if w is not a commutator word, we can use [6,Proposition 5.10] in order to see, that the nonsolvable length λ(G) is bounded by log 2 (|w|). Therefore, λ(G) can possibly be bounded in general by a function in O(log(ν(G))).…”

“…Hence K ∩ L ≤ F , and this implies that KF/F centralizes LF/F = S 1 (G)/F . On the other hand S 1 (G)/F has trivial centralizer in G/F (by[6, Lemma 2.4]). It follows that K ≤ F .…”

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

“…Our bound on the nonsolvable length can probably be improved: At least when $$w$$ is not a commutator word, we can use [6, Proposition 5.10] in order to see that the nonsolvable length $$\lambda (G)$$ is bounded by $$\log _2(|w|)$$. This argument uses the fact that a law of the form $$x_1^n\,\tilde{w}$$ with $$\tilde{w}\in F_\infty ^{\prime }$$ always implies the law $$x_1^n$$.…”

“…The proof of Theorem C relies heavily on the existence of so called rarefied subgroups. In [6, Theorem 1.1], it has been proved that every finite group of nonsolvable length $$m$$ contains $$m$$ ‐rarefied subgroups. These are subgroups of the same nonsolvable length $$m$$, with a very restricted structure.…”

“…They have shown in [9, Theorem 1.1], that the nonsolvable length of any finite group $$G$$ is bounded by $$2L_2(G)+1$$, where $$L_2(G)$$ denotes the maximum of the 2‐lengths of the solvable subgroups of $$G$$. This bound was improved in [6, Theorem 5.2] to $$L_2(G)$$. In this context, Khukhro and Shumyatsky raised the following question [9, Problem 1.3].…”