A Note on the $P_3$-Sequenceability of Finite Groups

Abstract

Applying Glauberman’s $Z^{\ast }$-theorem, it is shown that every finite group $G$ is strongly $P_3$-sequenceable, i.e. there exists a sequencing $(x_1,\dots ,x_N)$ of the elements of $G\setminus \{1\}$, such that all products $x_i{x}_{i+1}{x}_{i+2}$ ($1\le i\le N-2$), $x_{N-1}{x}_N{x}_1$ and $x_N{x}_1{x}_2$ are nontrivially rewritable. This was conjectured by J. Nielsen in~[N].