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

An 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 lengths n > k , as the permutation of the points. Looking at Steiner triple systems we show that this holds for all n unless n | Cp ( n )| ⩽ 3, where Cp ( n ) is the set of cycles of length n of 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 with k finite, we show that these results generalize. © 1995 John Wiley & Sons, Inc.