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Steiner triple system and maps into group

Source: Schweitzer 2009

November 13, 2009
superior algebrasuperior algebra unsolved

Problem Statement

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