In combinatorial mathematics, a Dowling geometry, named after Thomas A. Dowling, is a matroid associated with a group. There is a Dowling geometry of each rank for each group. If the rank is at least 3, the Dowling geometry uniquely determines the group. Dowling geometries have a role in matroid theory as universal objects (Kahn and Kung, 1982); in that respect they are analogous to projective geometries, but based on groups instead of fields. A Dowling lattice is the geometric lattice of flats associated with a Dowling geometry. The lattice and the geometry are mathematically equivalent: knowing either one determines the other. Dowling lattices, and by implication Dowling geometries, were introduced by Dowling (1973a,b). A Dowling lattice or geometry of rank n of a group G is often denoted Qn(G). The original definitions 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 corresponding Dowling geometry is the set of 1-dimensional vector subspaces generated by the elements of E. In his second paper (1973b) Dowling gave an intrinsic definition of the rank-n Dowling lattice of any finite group G. Let S be the set {1,...,n}. A G-labelled set (T, α) is a set T together with a function α: T → G. Two G-labelled sets, (T, α) and (T, β), are equivalent if there is a group element, g, such that β = gα. An equivalence class is denoted [T, α]. A partial G-partition of S is a set γ = {[B1,α1], ..., [Bk,αk]} of equivalence classes of G-labelled sets such that B1, ..., Bk are nonempty subsets of S that are pairwise disjoint. (k may equal 0.) A partial G-partition γ is said to be ≤ another one, γ*, if every block of the second is a union of blocks of the first, and for each Bi contained in B*j, αi is equivalent to the restriction of α*j to domain Bi . This gives a partial ordering of the set of all partial G-partitions of S. The resulting partially ordered set is the Dowling lattice Qn(G). The definitions are valid even if F or G is infinite, though Dowling mentioned only finite fields and groups. Graphical