Classification of highly symmetrical translation loops of order 2 p, p prime

Mikhail Klin, Nimrod Kriger, Andrew Woldar

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


Using tools from algebraic graph theory, we prove that for every odd prime p there is just one proper loop of order 2p such that the automorphism group of the corresponding rank 5 association scheme contains a regular subgroup of order 4p2. Although our results were initially obtained on the basis of theoretical generalization of a tremendous number of computer algebra experiments, our final computer-free proof uses quite elementary arguments from group theory and combinatorics. In this paper we provide this computer-free proof, as well as discuss further suggested use of our methods.

Original languageEnglish
Pages (from-to)253-276
Number of pages24
JournalBeitrage zur Algebra und Geometrie
Issue number1
StatePublished - 1 Mar 2014


  • 3-net
  • 4-vertex condition
  • Association scheme
  • Cayley graph
  • Finite permutation group
  • G-loop
  • Highly symmetrical loop
  • Intercalate
  • Loop
  • Partial difference set
  • Primitive S-ring
  • Strongly regular graph
  • Translation loop
  • Transversal design

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology


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