Two point stabilisers of partition actions of linear groups

A partition of a kl-dimensional vector space V is a set of l subspaces each of dimension k such that their direct sum is the original space V. In this paper we show that, unless l=2, the action of a group such that on the set of partitions of V into l subspaces of dimension k is base two: there exist two partitions V and W such that .We will also show that, given any finite group G, there exist k, l and partitions V, W such that .These results complement work the author has done with partition actions of the symmetric groups.