Nonabelian sets with distinct \(k\)-sums
A subset \(S\) of a finite abelian group \(G\) is called an \(S_k\)-set provided that all sums of \(k\) elements selected from \(S\) are distinct. In the special case \(k = 2\), this condition is equivalent to the requirement that all nonzero differences \(d - e\) with \(d,e \in S\) are distinct; thus planar difference sets provide examples, and finding large \(S_2\)-sets may be viewed as a sort of packing problem for difference sets. In view of the multiplier theory for difference sets, one might ask whether there are \(S_k\)-sets admitting ``multipliers''. As the authors show by modifying a construction of Bose and Chowla, the answer to this question is positive: For every integer \(k \geq 2\) and every prime \(p \equiv 1 \pmod k\), there exists an \(S_k\)-set \(S\) of cardinality \(p\) in \(Z_m\), where \(m = p^k - 1\), such that \(S = pS\). The authors also consider a nonabelian version of \(S_k\)-sets and construct large sets \(S\) in certain nonabelian groups such that all \(k\)-letter words with letters from \(S\) are distinct.
scientific article; zbMATH DE number 825351
scientific article