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

TL;DR

This paper characterizes finite-by-nilpotent groups through bounds on the cardinality of sets of commutators, establishing a criterion involving the parameters k and n that relate to the group's structure.

Contribution

It provides a new characterization of finite-by-nilpotent groups using bounds on commutator sets, linking group structure to combinatorial properties.

Findings

A group G is finite-by-nilpotent iff there exist k,n with |a|_k < n for all a in G.

If |a|_k < n for all a, then _{k+1}(G) has finite (k,n)-bounded order.

The paper establishes a precise relationship between commutator bounds and group nilpotency.

Abstract

For a group G and an element a in G let |a|_k denote the cardinality of the set of commutators [a,x_1,...,x_k], where x_1,...,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 \gamma_{k+1}(G) has finite (k,n)-bounded order.