### Basic Research Project J1-9108

Semiregular elements in 2-closures of solvable groups

Project Title: Semiregular elements in 2-closures of solvable groups

PI: Dragan Marušič.

Project Code: J1-9108

Type of the Project: basic research project.

Funding Organization: Slovenian Research Agency (ARRS).

Research Field (ARRS): 1.01.00 - Natural Sciences and Mathematics / Mathematics.

Duration: 1. 7. 2018 - 30. 6. 2021.

Project Category: B.

Yearly Range: 0.86 FTE (1.468 research hours).

Sicris profile of the project is avaliable __here__.

Research Organizations:

University of Primorska, Andrej Marušič Institute,

University of Primorska, Faculty of Mathematics, Natural Sciences and Information Technologies,

University of Ljubljana, Faculty of Education.

Project Members:

Kovacs Istvan (ARRS code: 25997) |

Dobson Edward Tauscher (ARRS code: 34109) Jajcay Robert (ARRS code: 37724) |

Malnič Aleksander (ARRS code: 02507) |

Marušič Dragan (leader) (ARRS code: 02887) |

Kuzman Boštjan (ARRS code: 23501) Šparl Primož (ARRS code: 23341) |

Woodroofe Russell Stephen (ARRS code: 50355) |

Pasalic Enes (ARRS code: 27777) |

Filipovski Slobodan (ARRS code: 37715) |

Ramos Rivera Alejandra (ARRS code: 37541) |

Cepak Nastja (ARRS code 37552) |

Abstract:

Let G be a permutation group on a finite set V. A non-identity element of G is semiregular if the subgroup of G generated by this element has all orbits of equal length. It is known that every finite transitive permutation group contains a fixed-point-free element of prime power order, but not necessarily a fixed-point-free element of prime order, that is, a semiregular element of prime order. A permutation group with no semiregular elements of prime order is sometimes called elusive, which is equivalent to requiring that it does not contain semiregular elements at all.

One would expect “nice” combinatorial objects, for example graphs, to have non-elusive automorphism groups. Indeed, the problem first arose in a graph-theoretic context in 1981 when the PI asked if every vertex-transitive digraph has a semiregular automorphism. The now commonly accepted, and more general, version of this question (known as the Polycirculant conjecture) involves the whole class of 2-closed transitive groups and is due to Klin.

Usually, when dealing with problems in algebraic graph theory, and more specifically transitive group actions on graphs, the hardest part is the non-solvable case, with the solvable case allowing at least some partial results. With the semiregularity problem, however, the situation is completely reversed. It is the class of solvable groups that presents the main obstacle to obtaining a complete solution. The proposed project aims to make further steps towards complete solution of the semiregularity problem by giving a special emphasis to transitive solvable groups. On the other hand, semiregular automorphisms will be used to contribute to other active topics of research in algebraic graph theory and beyond. In particular, the proposed project will also address a connection that went by unnoticed and lies at the intersection of discrete mathematics and the theory of computation: given two arrays of elements in a group, decide whether there exists an element of the group that simultaneously conjugates the two arrays. The problem is known as the simultaneous conjugacy problem. One of the goals of the proposed project is to develop efficient algorithms for solving the simultaneous conjugacy problem in the symmetric group, and to find non-trivial lower bounds for this problem. It is natural to consider this problem in the framework of the proposed project as the Sridhar’s algorithm for solving the simultaneous conjugacy problem does not work in the case when each permutation is semiregular.