On the existence of block-transitive Steiner designs with \(k\) divides \(v\). (English) Zbl 07841640
A Steiner \(t\) design \(t\)-\((v, k, 1)\) is a pair \((X, \mathcal B)\) where \(X\) is a set of \(v\) points, and \(\mathcal B\) is a collection of size \(k\) subsets of \(X\) (called blocks) such that any \(t\) points appear together in exactly one block. The classification of all block transitive Steiner \(t\) designs is a very interesting topic. The authors prove that there are no block-transitive \(4\)-\((v, k, 1)\) designs and \(6\)-\((v, k, 1)\) designs with \(k\) divides \(v\).
Reviewer: Zhaoping Meng (Jinan)
MSC:
| 05B05 | Combinatorial aspects of block designs |
| 20B25 | Finite automorphism groups of algebraic, geometric, or combinatorial structures |
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