A combinatorial approach to transitive extensions of generously unitransitive permutation groups

M. H. Klin, D. M. Mesner, A. J. Woldar

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Abstract

Motivated by symmetric association schemes (which are known to approximate generously unitransitive group actions), we formulate combinatorial approximations to transitive extensions of generously unitransitive permutation groups. Specifically, the notions of compatible and coherent partitions are suggested and investigated in terms of the orbits of an ambient group (H,Ω) on the κ-subsets of X, k=2,3,4. We apply these ideas to investigate transitive extensions of the automorphism groups of the classical Johnson and Hamming schemes. In the latter case, we further provide algorithmic details and computer-generated data for the particular series of Hamming schemes H(m,3), m≥2. Finally, our approach is compared to the concept of a symmetric association scheme on triples in the sense of Mesner and Bhattacharya.

Original languageEnglish
Pages (from-to)369-391
Number of pages23
JournalJournal of Combinatorial Designs
Volume18
Issue number5
DOIs
StatePublished - 1 Jan 2010

Keywords

  • 3-homogeneous graph
  • Association scheme
  • Association scheme on triples
  • Generously unitransitive
  • Hamming scheme
  • Permutation group
  • Transitive extension
  • Triangular graph
  • Triple design
  • κ-orbit

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