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Cycle structures of automorphisms of 2‐(v,k,1) designs
Author(s) -
Webb Bridget S.
Publication year - 1995
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.3180030504
Subject(s) - mathematics , automorphism , combinatorics , permutation (music) , automorphism group , steiner system , permutation group , set (abstract data type) , discrete mathematics , action (physics) , computer science , physics , quantum mechanics , acoustics , programming language
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 n | Cp ( n )| ⩽ 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. © 1995 John Wiley & Sons, Inc.