A Dowling lattice or geometry of rank n of a group G is often denoted Qn(G).
In his first paper (1973a) Dowling defined the rank-n Dowling lattice of the multiplicative group of a finite field F. It is the set of all those subspaces of the vector space Fn that are generated by subsets of the set E that consists of vectors with at most two nonzero coordinates.
The definitions are valid even if F or G is infinite, though Dowling mentioned only finite fields and groups.
We give the slightly simpler (but essentially equivalent) graphical definition of Zaslavsky (1991), expressed in terms of gain graphs.
This gives a graph which is called GKno (note the raised circle).
To define the Dowling geometry, we specify the circuits (minimal dependent sets).
One reason for interest in Dowling lattices is that the characteristic polynomial is very simple.