Groups with finitely many long commutators of maximal order

Abstract

Given a group $G$ and elements $x_1,x_2,\dots ,x_{\ell }\in G$, the commutator of the form $[x_1,x_2,\dots ,x_{\ell }]$ is called a commutator of length $\ell$. The present paper deals with groups having only finitely many commutators of length $\ell$ of maximal order. We establish the following results. Let $G$ be a residually finite group with finitely many commutators of length $\ell$ of maximal order. Then $G$ contains a subgroup $M$ of finite index such that ${\gamma }_{\ell }(M)=1$. Moreover, if $G$ is finitely generated, then ${\gamma }_{\ell }(G)$ is finite. Let $\ell ,m,n,r$ be positive integers and $G$ an $r$-generator group with at most $m$ commutators of length $\ell$ of maximal order $n$. Suppose that either $n$ is a prime power, or $n=p^{\alpha }{q}^{\beta }$, where $p$ and $q$ are odd primes, or $G$ is nilpotent. Then ${\gamma }_{\ell }(G)$ is finite of $(m,\ell ,r)$-bounded order and there is a subgroup $M\le G$ of $(m,\ell ,r)$-bounded index such that ${\gamma }_{\ell }(M)=1$.