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Resolutions of PG(5, 2) with point‐cyclic automorphism group
Author(s) -
Sarmiento Jumela
Publication year - 2000
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/(sici)1520-6610(2000)8:1<2::aid-jcd2>3.0.co;2-h
Subject(s) - mathematics , combinatorics , automorphism , automorphism group , disjoint sets , point (geometry) , invariant (physics) , group (periodic table) , cyclic group , discrete mathematics , geometry , chemistry , organic chemistry , abelian group , mathematical physics
A t ‐(υ, k , λ) design is a set of υ points together with a collection of its k ‐subsets called blocks so that all subsets of t points are contained in exactly λ blocks. The d ‐dimensional projective geometry over GF(q), PG( d, q ), is a 2‐( q d + q d −1 + … + q + 1, q + 1, 1) design when we take its points as the points of the design and its lines as the blocks of the design. A 2‐(υ, k, 1) design is said to be resolvable if the blocks can be partitioned as ℛ = { R 1 , R 2 , …, R s }, where s = (υ − 1)/( k −1) and each R i consists of υ/ k disjoint blocks. If a resolvable design has an automorphism σ which acts as a cycle of length υ on the points and ℛ σ = ℛ , then the design is said to be point‐cyclically resolvable . The design associated with PG(5, 2) is known to be resolvable and in this paper, it is shown to be point‐cyclically resolvable by enumerating all inequivalent resolutions which are invariant under a cyclic automorphism group G = 〈σ〉 where σ is a cycle of length 63. These resolutions are the only resolutions which admit a point‐transitive automorphism group. Furthermore, some necessary conditions for the point‐cyclic resolvability of 2‐(υ, k , 1) designs are also given. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 2–14, 2000