Composition series in groups and the structure of slim semimodular lattices
Tóm tắt
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.
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