Nilpotence in finitary skew linear groups

Abstract

We study nilpotence properties (upper central series, Engel elements, central heights, etc.) of groups of finitary linear transformations of a vector space over a division ring that is locally finite-dimensional over its centre. Not surprisingly properties of skew linear groups over these divisions rings do not usually extend to this much more general setting. However, not all is lost. Although the group itself may not have the property, frequently it has a local system of normal subgroups with the property. In many ways this is a surprisingly strong conclusion. Most of the difficulties involve unipotent elements. This paper really concerns the development of bounds for unipotent skew linear groups of various kinds that are independent of the degree and hence are potentially meaningful in the finitary, infinite-dimensional situation. Some of these give new insights even in the ordinary linear case. Once this is done the nilpotence results hinted at above are then easy corollaries.