Classification of Groups Which Admit a Finite Number of Distinct Right-Orders

Tararin has shown that a non-Abelian group G admits a nonzero finite number of distinct right-orders if and only if G is equipped with a Tararin-type series of some length n. Further, a group which has a Tararin-type series of length n admits 2 ⁿ right-orders. It is known that a group has two right-orders if and only if it is torsionfree Abelian of rank 1. Here we completely classify the groups which admit either four or eight right-orders.