Orbits on cycles of automorphisms

Abstract

Block's Lemma states that an automorphism group G of a (finite) maximal rank incidence structure acts with at least as many orbits on the blocks as on the points. We look at the action of G on the set C_i(G) of all i-cycles that occur in the cycle decomposition of some element of G, and show that cyclic and abelian automorphism groups act with at least as many orbits on C_i(G^B) as on C_i(G^p). We give examples of maximal rank structures with more orbits on the point 2-cycles than on the block 2-cycles, showing that Block's Lemma cannot be generalized to these actions on cycles.