On the full automorphism group of a Hamiltonian cycle system of odd order

TL;DR

This paper characterizes the possible full automorphism groups of Hamiltonian cycle systems of odd order, establishing necessary and sufficient conditions involving group properties, with some exceptions for non-solvable binary groups.

Contribution

It provides a complete characterization of automorphism groups for Hamiltonian cycle systems of odd order, identifying specific group classes that can occur.

Findings

Automorphism groups must have odd order, be binary, or be AGL(1; p) with p prime.

The necessary condition is also sufficient, except possibly for non-solvable binary groups.

The paper narrows down the group structures compatible with Hamiltonian cycle systems.

Abstract

It is shown that a necessary condition for an abstract group G to be the full automorphism group of a Hamiltonian cycle system is that G has odd order or it is either binary, or the affine linear group AGL(1; p) with p prime. We show that this condition is also sufficient except possibly for the class of non-solvable binary groups.