Groups with many rewritable products

Abstract:

For any integer $n \geq 2$, denote by ${R_n}$ the class of groups $G$ in which every infinite subset $X$ contains $n$ elements ${x_1}, \ldots ,{x_n}$ such that the product ${x_1} \ldots {x_n} = {x_{\sigma (1)}} \cdots {x_{\sigma (n)}}$ for some permutation $\sigma \ne 1$. The case $n = 2$ was studied by B. H. Neumann who proved that ${R_2}$ is precisely the class of centre-by-finite groups. Here we show that $G \in {R_n}$ for some $n$ if and only if $G$ contains an FC-subgroup $F$ of finite index such that the exponent of $F/Z(F)$ is finite, where $Z(F)$ denotes the centre of $F$.