Composition Series in Groups and the Structure of Slim Semimodular Lattices

Let ~ H and ~ K be finite composition series of a group G. The intersections Hi ∩ Kj of their members form a lattice CSL( ~ H, ~ K) under set inclusion. Improving the Jordan-Hölder theorem, G. Grätzer, J.B. Nation and the present authors have recently shown that ~ H and ~ K determine a unique permutation π such that, for all i, the i-th factor of ~ H is “down-and-up projective” to the π(i)-th factor of ~ K. Equivalent definitions of π were earlier given by R.P. Stanley and H. Abels. We prove that π determines the lattice CSL( ~ H, ~ K). More generally, we describe slim semimodular lattices, up to isomorphism, by permutations, up to an equivalence relation called “sectionally inverted or equal”. As a consequence, we prove that the abstract class of all CSL( ~ H, ~ K) coincides with the class of duals of all slim semimodular lattices.