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\begin{document}$$\overrightarrow{H}$$\end{document}and \documentclass[12pt]{minimal}
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\begin{document}$$\overrightarrow{K}$$\end{document} be finite composition series of a group G. The intersections Hi ∩ Kj of their members form a lattice CSL(\documentclass[12pt]{minimal}
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\begin{document}$$\overrightarrow{H}$$\end{document}, \documentclass[12pt]{minimal}
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\begin{document}$$\overrightarrow{K}$$\end{document}) under set inclusion. Improving the Jordan-Hölder theorem, G. Grätzer, J. B. Nation and the present authors have recently shown that \documentclass[12pt]{minimal}
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\begin{document}$$\overrightarrow{H}$$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$$\overrightarrow{K}$$\end{document} determine a unique permutation π such that, for all i, the i-th factor of \documentclass[12pt]{minimal}
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\begin{document}$$\overrightarrow{H}$$\end{document}is “down-and-up projective”to the π(i)-th factor of \documentclass[12pt]{minimal}
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\begin{document}$$\overrightarrow{K}$$\end{document}. Equivalent definitions of π were earlier given by R. P. Stanley and H. Abels. We prove that π determines the lattice CSL(\documentclass[12pt]{minimal}
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\begin{document}$$\overrightarrow{H}$$\end{document}, \documentclass[12pt]{minimal}
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\begin{document}$$\overrightarrow{K}$$\end{document}). 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(\documentclass[12pt]{minimal}
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\begin{document}$$\overrightarrow{H}$$\end{document}, \documentclass[12pt]{minimal}
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\begin{document}$$\overrightarrow{K}$$\end{document}) coincides with the class of duals of all slim semimodular lattices.
