A Note on the \(P_3\)-Sequenceability of Finite Groups

Applying Glauberman’s \(Z^*\)-theorem, it is shown that every finite group \(G\) is strongly \(P_3\)-sequenceable, i.e. there exists a sequencing \((x_1,\ldots,x_{N})\) of the elements of \(G\setminus\{1\}\), such that all products \(x_ix_{i+1}x_{i+2}\) (\(1\leq i\leq N-2\)), \(x_{N-1}x_{N}x_{1}\) and \(x_Nx_{1}x_2\) are nontrivially rewritable. This was conjectured by J. Nielsen in~[N].