Orbits on cycles of automorphisms - Mendelsohn, Eric; Webb, Bridget S. (Utilitas Mathematica, 2000)
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 Ci(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 Ci(GB) as on Ci(Gp). 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.