[Paper Review] Nonsoluble and non-p-soluble length of finite groups

E. I. Khukhro, Pavel Shumyatsky|arXiv (Cornell University)|Oct 9, 2013

Finite Group Theory Research11 references4 citations

TL;DR

This paper establishes upper bounds for the nonsoluble length λ(G) and non-p-soluble length λₚ(G) of finite groups in terms of the 2-length and p-length of their soluble subgroups, respectively. It proves that λ(G) ≤ 2L₂ + 1 and λₚ(G) ≤ Lₚ for p ≠ 2, with an application correcting a flaw in a prior proof concerning verbal subgroups in residually finite groups.

ABSTRACT

Every finite group $G$ has a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. We define the nonsoluble length $λ(G)$ as the minimum number of nonsoluble factors in a series of this kind. Upper bounds for $λ(G)$ appear in the study of various problems on finite, residually finite, and profinite groups. We prove that $λ(G)$ is bounded in terms of the maximum $2$-length of soluble subgroups of $G$, and that $λ(G)$ is bounded by the maximum Fitting height of soluble subgroups. For an odd prime $p$, the non-$p$-soluble length $λ_p(G)$ is introduced, and it is proved that $λ_p(G)$ does not exceed the maximum $p$-length of $p$-soluble subgroups. We conjecture that for a given prime $p$ and a given proper group variety ${\frak V}$ the non-$p$-soluble length $λ_p(G)$ of finite groups $G$ whose Sylow $p$-subgroups belong to ${\frak V}$ is bounded. In this paper we prove this conjecture for any variety that is a product of several soluble varieties and varieties of finite exponent.

Motivation & Objective

• To establish upper bounds for the nonsoluble length λ(G) of finite groups based on structural properties of their soluble subgroups.
• To introduce and bound the non-p-soluble length λₚ(G) for odd primes p, analogous to the nonsoluble length.
• To resolve a conjecture on bounded non-p-soluble length for finite groups whose Sylow p-subgroups lie in a product of soluble and finite exponent varieties.
• To correct a critical error in the proof of Proposition 3.3 in Khukhro & Shumyatsky (2012) concerning verbal subgroups in residually finite groups.
• To provide a revised, valid proof of the boundedness of verbal subgroup exponents using Fitting height and nonsoluble length bounds.

Proposed method

• Define the nonsoluble length λ(G) as the minimal number of nonsoluble factors in a normal series of G with soluble or direct product of nonabelian simple group factors.
• Introduce the non-p-soluble length λₚ(G) by replacing 'soluble' with 'p-soluble' in the series definition.
• Use the maximum 2-length of soluble subgroups to bound λ(G) via a recursive construction of normal series.
• Apply the maximum p-length of p-soluble subgroups to bound λₚ(G), with a refined proof for p ≠ 2.
• Leverage Corollary 1.2, which bounds λ(G) by the maximum Fitting height of soluble subgroups, to correct flawed arguments in prior proofs.
• Apply Jones' result on generation of group varieties by infinite families of finite simple groups to bound the order of the nonsoluble part of the series.

Experimental results

Research questions

• RQ1 Can the nonsoluble length λ(G) of a finite group be bounded in terms of the 2-length of its soluble subgroups?
• RQ2 Is the non-p-soluble length λₚ(G) bounded by the p-length of p-soluble subgroups for odd primes p?
• RQ3 Does the non-p-soluble length λₚ(G) remain bounded for finite groups whose Sylow p-subgroups lie in a product of soluble and finite exponent varieties?
• RQ4 Can the error in Lemma 2.5 of Khukhro & Shumyatsky (2012) be corrected using the new bounds on nonsoluble length?
• RQ5 Can the boundedness of verbal subgroup exponents in residually finite groups be re-proven without relying on the flawed lemma?

Key findings

• The nonsoluble length λ(G) of a finite group G is bounded by 2L₂ + 1, where L₂ is the maximum 2-length of its soluble subgroups.
• For p ≠ 2, the non-p-soluble length λₚ(G) is bounded by the maximum p-length of p-soluble subgroups of G.
• The nonsoluble length λ(G) is bounded by the maximum Fitting height of soluble subgroups of G.
• The conjecture that λₚ(G) is bounded for finite groups with Sylow p-subgroups in a product of soluble and finite exponent varieties is proven true.
• The flawed proof in Khukhro & Shumyatsky (2012) regarding verbal subgroups is corrected using Corollary 1.2 and the new bounds on nonsoluble length.
• The revised proof establishes that the order of a finite group G is bounded in terms of k, m, n when all products of 896 δₖ-commutators have order dividing n and generators satisfy order constraints.