On groups with the Lohse property

A finite group has the Lohse property if the rows and columns of its Cayley table can be arranged in such a way that entries in diagonally adjacent positions are different. In this paper we determine the groups with the Lohse property completely: all finite groups except the groups of order less than or equal to four and the quaternion group have the Lohse property. The decisive properties are that the group has a generating system, which is closed under taking inverses, of size less than half the order of the group, and the existence of a Hamilton cycle in the Cayley graph for certain generating systems of the group.