A set system (S,L) is called a Steiner triple system, if L=∅, any pair x,y∈S, x=y of points lie on a unique line ℓ∈L, and every line ℓ∈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=y. Let A be a group whose factor by its center C(A) is of prime power order. Let f,h:S→A be maps, such that C(A) contains the range of f, and the range of h generates A.
Show, that if
f(x) \equal{} h(x)h(y)h(x)h(xy)
holds for all pairs x=y of points, then A is commutative, and there exists an element k∈A, such that f(x) \equal{} kh(x) for all x∈S. superior algebrasuperior algebra unsolved