A class of geometric lattices based on finite groups

For any finite group G and positive integer n a finite geometric lattice Qn(G) of rank n, the lattice of partial G-partitions, is constructed. Let Pn+1 be the lattice of partitions of an (n+1)-set. There exists a surjection π: Qn(G) → Pn+1, and an injection t: Pn+1 → Qn+1(G), each of which preserves order and rank. When G is the trivial group, π=t−1 reduces to an isomorphism. The interval structure, Möbius function, and characteristic polynomial of Qn(G) are determined, and Stirling-like identities for the Whitney numbers obtained. The existence of a Boolean sublattice of modular elements in Qn(G) is established, implying that Qn(G) is supersolvable. It is further shown that non-isomorphic groups give non-isomorphic lattices, and the representation problem is solved completely: Qn(G) is representable over a field when n≥3 if and only if G is isomorphic to a subgroup of the multiplicative group of the field. Consequently, Qn(G) is representable over no field iff G is noncyclic, and, if G is cyclic of order m, then Qn(G) is representable over (a) every field iff m=1, (b) a finite field of order q iff m divides q−1, (c) the rational or real fields iff m=1 or 2, and (d) the complex field for all m