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Symmetric (41,16,6)‐designs with a nontrivial automorphism of odd order
Author(s) -
Spence Edward
Publication year - 1993
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.3180010303
Subject(s) - mathematics , isomorphism (crystallography) , combinatorics , automorphism , order (exchange) , prime (order theory) , dual (grammatical number) , combinatorial design , duality (order theory) , automorphism group , discrete mathematics , art , chemistry , literature , finance , crystal structure , economics , crystallography
If a symmetric (41,16,6)‐design has an automorphism σ of odd prime order q then q = 3 or 5. In the case q = 5 we determine all such designs and find a total of 419 nonisomorphic ones, of which 15 are self‐dual. When q = 3 a combinatorial explosion occurs and the complete classification becomes impracticable. However, we give a characterization in the particular case when σ has order 3 and fixes 11 points, and find that there are 3,076 nonisomorphic designs with this property, all of them being non self‐dual. The other remaining possibility is that σ, of order 3, fixes 5 points. In this case there are 960 orbit matrices (up to isomorphism and duality) and all but one of them yield designs. Here an incomplete investigation shows that in total there are at least 112,000 nonisomorphic designs. © 1993 John Wiley & Sons, Inc.