Periodic-by-Nilpotent Linear Groups

Abstract

Let $G$ be a linear group of (finite) degree $n$ and characteristic $p\ge 0$. Suppose that for every infinite subset $X$ of $G$ there exist distinct elements $x$ and $y$ of $X$ with $⟨x,x^y⟩$ periodic-by-nilpotent. Then $G$ has a periodic normal subgroup $T$ such that if $p>0$ then $G/T$ is torsion-free abelian and if $p=0$ then $G/T$ is torsion-free nilpotent of class at most $\max 1,n-1$ and is isomorphic to a linear group of degree $n$ and characteristic zero. We also discuss the structure of periodic-by-nilpotent linear groups.