کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
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6423433 | 1342375 | 2012 | 14 صفحه PDF | دانلود رایگان |
Let Hâ and Kâ be finite composition series of length h in a group G. The intersections of their members form a lattice CSL(Hâ,Kâ) under set inclusion. Our main result determines the number N(h) of (isomorphism classes) of these lattices recursively. We also show that this number is asymptotically h!/2. If the members of Hâ and Kâ are considered constants, then there are exactly h! such lattices.Based on recent results of Czédli and Schmidt, first we reduce the problem to lattice theory, concluding that the duals of the lattices CSL(Hâ,Kâ) are exactly the so-called slim semimodular lattices, which can be described by permutations. Hence the results on h! and h!/2 follow by simple combinatorial considerations. The combinatorial argument proving the main result is based on Czédli's earlier description of indecomposable slim semimodular lattices by matrices.
Journal: Discrete Mathematics - Volume 312, Issue 24, 28 December 2012, Pages 3523-3536