Finite-by-nilpotent groups and a variation of the BFC-theorem

2021/10/21 by Pavel Shumyatsky, Shumyatsky, Pavel
Computer Science · Mathematics · #Advanced Graph Theory Research #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.2110.11379

openalex publication_date 2021/10/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a group G and an element a in G let |a|_k denote the cardinality of the set of commutators [a,x₁,...,x_k], where x₁,...,x_k range over G. The main result of the paper states that a group G is finite-by-nilpotent if and only if there are positive integers k and n such that |x|_k < n for every x in G. More precisely, if |x|_k < n for every x in G then γ_k+1(G) has finite (k,n)-bounded order.

Related

• On finite-by-nilpotent groups
• On finite groups with bounded conjugacy classes of commutators
• A note on FC-nilpotency
• BFC-theorems for higher commutator subgroups
• A nilpotency criterion for finite groups