Dowling Geometry - The Original Definitions

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 F n 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 . A partial G-partition of S is a set γ = {, ..., } of equivalence classes of G-labelled sets such that B₁, ..., B_k 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 B_i contained in B*_j, α_i is equivalent to the restriction of α*_j to domain B_i .

This gives a partial ordering of the set of all partial G-partitions of S. The resulting partially ordered set is the Dowling lattice Q_n(G).

The definitions are valid even if F or G is infinite, though Dowling mentioned only finite fields and groups.