A permutational property of groups
If \(n\) is an integer greater than 1, a group \(G\) is said to have the property \({\mathcal P}_ n\) if every ordered product of \(n\) elements of \(G\) can be rewritten in at least one way, i.e. to each n-tuple \((x_ 1,x_ 2,...,x_ n)\) with \(x_ i\in G\), there corresponds a nontrivial element \(\sigma\) of the symmetric group \(S_ n\) such that \(x_ 1x_ 2...x_ n=x_{\sigma(1)}x_{\sigma(2)}...x_{\sigma(n)}.\) Obviously \({\mathcal P}_ 2\) is commutativity, while \({\mathcal P}_ 3\), \({\mathcal P}_ 4\), etc. are successively weaker properties. A group has the property \({\mathcal P}\) if is satisfies \({\mathcal P}_ n\) for some \(n>1\). A complete description is given of groups with the property \({\mathcal P}\); these are precisely the groups which are finite-by-abelian-by-finite.
scientific article; zbMATH DE number 3865574