The finite Steinhaus problem asks whether every finite set A in the plane with |A| = n > 1 there cannot exist a set S in the plane such that the intersection of pi(A) and S has cardinality 1 for every isometric copy pi(A) of A. It is easy to see that this holds for n = 2 and n = 3, and a recent theorem of Xuan (extending results of Miller-Weiss and Gao) shows the result for n = 4. We make some connections between this problem and certain algebraic concepts, and show the result for n = 5 and n = 7. This is joint work with Devon Henkis and Steve Jackson.