On some problems regarding LCM-groups

Let G be a finite group and denote by o(g) the order of an element g∈G. We say that G is an LCM-group if o(xny) is a divisor of the least common multiple of o(xn) and o(y) for all x,y∈G and n∈N. This paper extends some existing results on LCM-groups, such as the structure of a minimal non-LCM-group, and establishes criteria for G to be an LCM-group or a nilpotent group. We also prove that, in general, a minimum cover of a finite set of LCM-groups is not an LCM-group, and we answer two questions posed by M. Amiri.