For each prime $p$ we show the existence of a partial Steinhaus set in the plane for the prime $p$ which can be obtained from the points $\{( i/p,j/p) : 0 ≤ i,j < p\}$ by translating by integer amounts only in the horizontal direction. We will also discuss the finite Steinhaus problem which asks whether for every finite set $A$ contained in $\mathbb{R}^2$ with $|A| = n > 1$ there cannot exist a set $S$ in $\mathbb{R}^2$ such that $|\pi(A)\cap S|=1$ for every isometric copy $\pi(A)$ of $A$. We make some connections between these problems and certain algebraic concepts.