Cycle structures of automorphisms of 2‐(v,k,1) designs

摘要

AbstractAn automorphism of a 2−(v,k, 1) design acts as a permutation of the points and as another of the blocks. We show that the permutation of the blocks has at least as many cycles, of lengthsn>k, as the permutation of the points. Looking at Steiner triple systems we show that this holds for allnunlessnCp(n) ⩽ 3, whereCp(n) is the set of cycles of lengthnof the automorphism in its action on the points. Examples of Steiner triple systems for each of these exceptions are given. Considering designs with infinitely many points, but withkfinite, we show that these results generalize. © 1995 John WileySons,