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Cycle structures of automorphisms of 2-(v,k,1) designs

✍ Scribed by Bridget S. Webb

Publisher: John Wiley and Sons
Year: 1995
Tongue: English
Weight: 460 KB
Volume: 3
Category: Article
ISSN: 1063-8539
DOI: 10.1002/jcd.3180030504

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✦ Synopsis

An automorphism of a 2-(v, k, 1) design acts as a permutation of the points and as another of the blocks. We show that the permutation of the blocks has at least as many cycles, of lengths n > k, as the permutation of the points. Looking at Steiner triple systems we show that this holds for all n unless nlCp(n)l 5 3, where Cp(n) is the set of cycles of length n of the automorphism in its action on the points. Examples of Steiner triple systems for each of these exceptions are given. Considering designs with infinitely many points, but with k finite, we show that these results generalize.

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