On elementary domains of partial projective representations of groups

We characterize the finite groups containing only elementary domains of factor sets of partial projective representations. A condition for a finite subset $A$ of a group $G,$ which contains the unity of the group, to induce an elementary partial representation of $G$ whose (idempotent) factor set is total is given. Finally, we characterize the elementary partial representation of abelian groups of degrees $\le 4$ with total factor set.