# On finite-by-nilpotent groups

## TL;DR

This paper proves that in groups where conjugacy classes of certain commutator values are uniformly bounded, the next lower central subgroup is finite with a bound depending on these parameters.

## Contribution

It generalizes Neumann's theorem by establishing finiteness of higher lower central subgroups under bounded conjugacy conditions.

## Key findings

- The (n+1)th lower central subgroup has finite order bounded by parameters m and n.
- Generalizes classical results on BFC-groups to broader classes of groups.
- Provides bounds on the size of certain subgroups based on conjugacy class sizes.

## Abstract

Let $\gamma_n=[x_1,\dots,x_n]$ be the $n$th lower central word. Denote by $X_n$ the set of $\gamma_n$-values in a group $G$ and suppose that there is a number $m$ such that $|g^{X_n}|\leq m$ for each $g\in G$. We prove that $\gamma_{n+1}(G)$ has finite $(m,n)$-bounded order. This generalizes the much celebrated theorem of B. H. Neumann that says that the commutator subgroup of a BFC-group is finite.