The Cross-Intersecting Family of Certain Permutation Groups

Abstract

Two subsets X and Y of a permutation group G acting on Ω are cross-intersecting if for every x∈X and every y∈Y there exists some point α∈Ω such that αx=αy. Based on several observations made on the cross-independent version of Hoffman’s theorem, we characterize in this paper the cross-intersecting families of certain permutation groups. Our proof uses a Cayley graph on a permutation subgroup with respect to the derangement. By carefully analyzing the cross-independent version of Hoffman’s theorem, we obtain a useful theorem to consider cross-intersecting subsets of certain kinds of permutation subgroups, such as PGL(2,q), PSL(2,q) and Sn.