Orbits of infinite block designs (Q1357756)

This article investigates automorphism groups of simple balanced incomplete block designs with \(v\) infinite and \(k\) finite. If such a group has \(N\) block and \(n\) point orbits then it is possible to have \(N<n\). The author gives bounds for these numbers showing, in particular, that \(N\geq n\) provided \(n\geq k(k-1)\). For the case \(k=3\) this is strengthened to \(N\geq n\) without restriction. The author also investigates block-transitive designs, giving bounds for the point-rank which are known in the case that the group is also point-transitive.