Groups synchronizing a transformation of non-uniform kernel

  • Authors:

  • João Araújo;Wolfram Bentz;Peter J. Cameron

  • Affiliations:

  • Universidade Aberta and Centro de Álgebra, Universidade de Lisboa, Av. Gama Pinto, 2, 1649-003 Lisboa, Portugal;Centro de Álgebra, Universidade de Lisboa, Av. Gama Pinto, 2, 1649-003 Lisboa, Portugal;Department of Mathematics, School of Mathematical Sciences at Queen Mary University of London, United Kingdom

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2013

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Abstract

Suppose that a deterministic finite automata A=(Q,@S) is such that all but one letters from the alphabet @S act as permutations of the state set Q and the exceptional letter acts as a transformation with non-uniform kernel. Which properties of the permutation group G generated by the letters acting as permutations ensure that A becomes a synchronizing automaton under every possible choice of the exceptional letter (provided the exceptional letter acts as a transformation of non-uniform kernel)? Such permutation groups are called almost synchronizing. It is easy to see that an almost synchronizing group must be primitive; our conjecture is that every primitive group is almost synchronizing. Clearly every synchronizing group is almost synchronizing. In this paper we provide two different methods to find non-synchronizing, but almost synchronizing groups. The infinite families of examples provided by the two different methods have few overlaps. The paper closes with a number of open problems on group theory and combinatorics.