Composition series in groups and the structure of slim semimodular lattices
Let H and K be finite composition series of a group G. The intersections Hi fl Kj of their members form a lattice CSL(/f, K) under set inclusion. Improving the Jordan-Holder theorem, G. Gratzer, J. B. Nation and the present authors have recently shown that H and K determine a unique permutation 7r s...
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| Dokumentumtípus: | Cikk |
| Megjelent: |
Bolyai Institute, University of Szeged
Szeged
2013
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| Sorozat: | Acta scientiarum mathematicarum
79 No. 3-4 |
| Kulcsszavak: | Matematika |
| Tárgyszavak: | |
| Online Access: | http://acta.bibl.u-szeged.hu/32900 |
| Tartalmi kivonat: | Let H and K be finite composition series of a group G. The intersections Hi fl Kj of their members form a lattice CSL(/f, K) under set inclusion. Improving the Jordan-Holder theorem, G. Gratzer, J. B. Nation and the present authors have recently shown that H and K determine a unique permutation 7r such that, for all i, the i-th factor of H is "down-and-up projective" to the 7r(i)-th factor of K. Equivalent definitions of 7r were earlier given by R. P. Stanley and H. Abels. We prove that 7r determines the lattice CSL(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. |
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| Terjedelem/Fizikai jellemzők: | 369-390 |
| ISSN: | 0001-6969 |