SCHWEITZER 2009 Problem 6

A set system $(S,L)$ is called a Steiner triple system, if $L\neq\emptyset$ , any pair $x,y\in S$ , $x\neq y$ of points lie on a unique line $\ell\in L$ , and every line $\ell\in L$ contains exactly three points. Let $(S,L)$ be a Steiner triple system, and let us denote by $xy$ the thrid point on a line determined by the points $x\neq y$ . Let $A$ be a group whose factor by its center $C(A)$ is of prime power order. Let $f,h: S\to A$ be maps, such that $C(A)$ contains the range of $f$ , and the range of $h$ generates $A$ . Show, that if $$f(x) = h(x)h(y)h(x)h(xy)$$ holds for all pairs $x\neq y$ of points, then $A$ is commutative, and there exists an element $k\in A$ , such that $f(x) = kh(x)$ for all $x\in S$ .