Composition series in groups and the structure of slim semimodular lattices

Abstract

Let $$\overrightarrow{H}$$ and $$\overrightarrow{K}$$ be finite composition series of a group G. The intersections Hi ∩ Kj of their members form a lattice CSL( $$\overrightarrow{H}$$ , $$\overrightarrow{K}$$ ) under set inclusion. Improving the Jordan-Hölder theorem, G. Grätzer, J. B. Nation and the present authors have recently shown that $$\overrightarrow{H}$$ and $$\overrightarrow{K}$$ determine a unique permutation π such that, for all i, the i-th factor of $$\overrightarrow{H}$$ is “down-and-up projective”to the π(i)-th factor of $$\overrightarrow{K}$$ . Equivalent definitions of π were earlier given by R. P. Stanley and H. Abels. We prove that π determines the lattice CSL( $$\overrightarrow{H}$$ , $$\overrightarrow{K}$$ ). 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( $$\overrightarrow{H}$$ , $$\overrightarrow{K}$$ ) coincides with the class of duals of all slim semimodular lattices.