Nonsoluble Length Of Finite Groups with Commutators of Small Order

TL;DR

This paper investigates bounds on the non-p-soluble length of finite groups with Sylow p-subgroups in specific varieties, proving such bounds exist when p is odd and the variety constrains commutator orders.

Contribution

It establishes the existence of bounds for non-p-soluble length in finite groups with Sylow p-subgroups in certain varieties, specifically for odd primes and commutator word conditions.

Findings

Bound exists for non-p-soluble length when p is odd and Sylow p-subgroups satisfy the variety conditions.

Affirmative answer to the boundedness question in the specified case.

Results connect group structure with properties of commutator values.

Abstract

Let p be a prime. Every finite group G has a normal series each of whose quotients either is p-soluble or is a direct product of nonabelian simple groups of orders divisible by p. The non-p-soluble length of G is defined as the minimal number of non-p-soluble quotients in a series of this kind. We deal with the question whether, for a given prime p and a given proper group variety V, there is a bound for the non-p-soluble length of finite groups whose Sylow p-subgroups belong to V. Let the word w be a multilinear commutator. In the present paper we answer the question in the affirmative in the case where p is odd and the variety is the one of groups in which the w-values have orders dividing a fixed number.