How many ways can two composition series intersect?

Abstract

Let H→ and K→ be finite composition series of length h in a group G. The intersections of their members form a lattice CSL(H→,K→) under set inclusion. Our main result determines the number N(h) of (isomorphism classes) of these lattices recursively. We also show that this number is asymptotically h!/2. If the members of H→ and K→ are considered constants, then there are exactly h! such lattices.Based on recent results of Czédli and Schmidt, first we reduce the problem to lattice theory, concluding that the duals of the lattices CSL(H→,K→) are exactly the so-called slim semimodular lattices, which can be described by permutations. Hence the results on h! and h!/2 follow by simple combinatorial considerations. The combinatorial argument proving the main result is based on Czédli’s earlier description of indecomposable slim semimodular lattices by matrices.