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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.

On the Full Automorphism Group of a Hamiltonian Cycle System of Odd Order