Rewritable Sequencings of Groups

Abstract

A finite group is called $P_n$-sequenceable if its nonidentity elements can be listed $x_1,x_2,\dots ,x_k$ 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.