The finite Steinhaus problem asks whether for every finite subset $A$ of $\mathbb{R}^2$ with $|A|=n>1$ there cannot exist a set $S$ contained in $\mathbb{R}^2$ such that $|\pi(A)\cap S|=1$ for every isometric copy $\pi(A)$ of $A$. It is easy to see that holds for $n=2, 3$ 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 $n=5, 7$.