Groups with finitely many long commutators of maximal order

2025/04/14 by Heras, Iker de las, Di Concilio, Federico, Shumyatsky, Pavel
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2504.10377

Abstract

Given a group G and elements x₁,x₂,…, x_ℓ∈ G, the commutator of the form [x₁,x₂,…, x_ℓ] is called a commutator of length ℓ. The present paper deals with groups having only finitely many commutators of length ℓ of maximal order. We establish the following results. Let G be a residually finite group with finitely many commutators of length ℓ of maximal order. Then G contains a subgroup M of finite index such that γ_ℓ(M)=1. Moreover, if G is finitely generated, then γ_ℓ(G) is finite. Let ℓ,m,n,r be positive integers and G an r-generator group with at most m commutators of length ℓ of maximal order n. Suppose that either n is a prime power, or n=p^αq^β, where p and q are odd primes, or G is nilpotent. Then γ_ℓ(G) is finite of (m,ℓ,r)-bounded order and there is a subgroup M≤ G of (m,ℓ,r)-bounded index such that γ_ℓ(M)=1.

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