Rewritable Sequencings of Groups

A finite group is called \(P_n\)-sequenceable if its nonidentity elements can be listed \(x_1, x_2, \ldots, x_{k}\) such that the product \(x_i x_{i+1} \cdots x_{i+n-1}\) can be rewritten in at least one nontrivial way for all \(i\). It is shown that \(S_n, A_n, D_n\) are \(P_3\)-sequenceable, that every finite simple group is \(P_4\)-sequenceable, and that every finite group is \(P_5\)-sequenceable. It is conjectured that every finite group is \(P_3\)-sequenceable.