ACTA issues

Composition series in groups and the structure of slim semimodular lattices

Gábor Czédli, E. Tamás Schmidt

Acta Sci. Math. (Szeged) 79:3-4(2013), 369-390

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

AMS Subject Classification (1991): 06C10, 20E15

Keyword(s): composition series, Jordan--Hölder Theorem, group, slim lattice, semimodularity, planar lattice, permutation

Received May 8, 2013, and in revised form May 16, 2013. (Registered under 30/2013.)