Geometric groups of second order and related combinatorial structures.

In 1977, D. Betten defined a geometric group to be a permutation group (G,Ω) such that G=Aut(R) for some hypergraph R on Ω. In this paper, we extend Betten's notion of a geometric group to what we call a geometric group of second order. By definition, this is a permutation group for which G=Aut(R) for some set R={R1,R2,...,Rd} of hypergraphs on Ω. Our main focus will be on permutation groups that are geometric of second order but not geometric. Within this small class of groups one finds the projective groups PGL(2,8),PΓL(2,8) and the affine groups AGL(1,8),AΓL(1,8). Our investigations, which are based primarily on these four groups, lead us to consider some familiar combinatorial structures (eg, Fano plane and affine design) in a less familiar context (overlarge sets of Steiner systems). [ABSTRACT FROM AUTHOR]