### Customized Project N1-0062

Oddness degree of transitive permutation groups

Project Title: Oddness degree of transitive permutation groups.

PI: Dragan Marušič.

Project Code: N1-0062.

Type of the Project: customized project.

Funding Organization: Slovenian Research Agency (ARRS).

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

Duration: 1. 6. 2017 - 31. 5. 2020.

Project Category: B.

Yearly Range: 1.82 FTE (3099 research hours).

Sicris profile of the project is avaliable here.

Research Organization: University of Primorska, Andrej Marušič Institute.

Project Members:

Hujdurović Ademir (ARRS code: 32518) |

Kovacs Istvan (ARRS code: 25997) |

Kutnar Klavdija (ARRS code: 24997) Jajcay Robert (ARRS code: 37724) |

Malnič Aleksander (ARRS code: 02507) |

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

Miklavič Štefko (ARRS code: 21656) Požar Rok (ARRS code: 32026) |

Abstract:

A transitive group is called even if all of its elements act as even permutations, and it is said to be odd otherwise. The following question will be addressed:

The intersection of automorphism groups of all basic orbital graphs of a group H (acting on an appropriate coset space) is called the 2-closure H^(2) of H. The group H is said to be 2-closed whenever H^(2)=H. Is it possible to find a structural distinguishing of even groups in regard to their proximity to odd groups via appropriate group imbeddings? More precisely, assuming that H is even, is it possible to imbed it into an odd group using its collection of basic orbital (di)graphs?

- If H is not 2-closed, is the 2-closure H^(2) odd (that is, all basic orbital (di)graphs admit odd automorphisms)? If the answer is yes, H is said to be closure-odd.

- If H^(2) is even, does there exist at least one basic orbital (di)graph of H admitting an odd automorphism? If the answer is yes, H is said to be orbital-odd, and is said to be zero-odd (even-closed) otherwise.

The Core OddGroups Research Problem is: Determine which of these three possibilities, occurs for H.

Examples of each of the three possibilities exist. The group A5 acting on the Petersen graph is even and closure-odd. The group PSL(2,17) acting on the unique cubic arc-transitive graph of order 102 is even and zero-odd. There are also examples of even groups which are orbital-odd; in fact, groups with even 2-closures for which every orbital (di)graph has odd automorphisms. Such is the case with SL(2,16) of degree 51 acting on the coset space of a subgroup of order 80 and giving rise to basic orbital (di)graphs of two isomorphism classes (a Fermat graph and a union of 17 directed 3-cycles).