Finite central height in linear groups

AbstractJ.D. Dixon has characterized those pairs (n,F), where n is a positive integer and F a field, for which every locally nilpotent subgroup of GL(n,F) is nilpotent. He showed further that these pairs (n,F) have the stronger property that there is a bound on the nilpotency class of the nilpotent subgroups of GL(n,F). In this note we show that these pairs (n,F) have the still stronger property that every subgroup of GL(n,F) has finite bounded central height. Our main result generalizes to groups of automorphisms of Noetherian modules over commutative rings