A Partial Solution to a Question Raised by R. L. Graham

A group \((G, \cdot)\) with the property that, for a particular integer \(r > 0\), every \(r\)-set \(S\) of \(G\) possesses an ordering, \(s_1, s_2, \ldots, s_r\), such that the partial products \(s_1, s_1s_2, \ldots, s_1 s_2 \cdots s_r\) are all different, is called an \(r\)-set-sequenceable group. We solve the question as to which abelian groups are \(r\)-set-sequenceable for all \(r\), except that, for \(r = n – 1\), the question is reduced to that of determining which groups are \(R\)-sequenceable.