Intersecting families in the alternating group and direct product of symmetric groups

Abstract

Let Sn denote the symmetric group on [n] = {1,...,n}. A family I ⊆ Sn is intersecting if any two elements of I have at least one common entry. It is known that the only intersecting families of maximal size in Sn are the cosets of point stabilizers. We show that, under mild restrictions, analogous results hold for the alternating group and the direct product of symmetric groups.