Composition series in groups and the structure of slim semimodular lattices

TL;DR

This paper explores the structure of lattices formed by intersections of composition series in groups, establishing a connection with slim semimodular lattices and permutations, thus extending the Jordan-Hölder theorem.

Contribution

It characterizes the lattice CSL(H,K) using permutations and describes slim semimodular lattices up to an equivalence, linking group theory and lattice theory.

Findings

CSL(H,K) is determined by a unique permutation pi.

Slim semimodular lattices are classified by permutations up to an equivalence.

The class of CSL(H,K) coincides with duals of slim semimodular lattices.

Abstract

Let H and K be finite composition series of a group G. The intersections H_i\cap K_j of their members form a lattice CSL(H,\K) under set inclusion. Improving the Jordan-H\"older theorem, G. Gr\"atzer, J.B. Nation and the present authors have recently shown that H and K determine a unique permutation pi such that, for all i, the i-th factor of H is "down-and-up projective" to the pi(i)-th factor of K. Equivalent definitions of pi were earlier given by R.P. Stanley and H. Abels. We prove that pi determines the lattice CLS(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.