General bounds for permutability in finite groups (Q1824031)

For an integer \(n\geq 2\) we say that a group \(G\in {\mathbb{P}}_ n\) if \[ (\forall x_ 1,...,x_ n\in G) (\exists \sigma \neq 1\in S_ n) (x_ 1...x_ n=x_{1\sigma}...x_{n\sigma}). \] If \((\exists n)(G\in {\mathbb{P}}_ n)\), then per(G) denotes the least such n. Clearly \({\mathbb{P}}_ 2\) is the class of abelian groups, and results on finite groups in \({\mathbb{P}}_ 3\) and \({\mathbb{P}}_ 4\) have been obtained by the author and others, see, for example \textit{M. Bianchi}, \textit{R. Brandl} and \textit{A. Gillio Berta Mauri} [Arch. Math. 48, 281-285 (1987; Zbl 0623.20022)]. The main results in this paper are: Theorem A. There exists a function f(n) with the property: if \(G\in {\mathbb{P}}_ n\) and G is finite, then \(| G/F(G)| \leq f(n)\). Theorem B. For every \(\epsilon >0\) there exist only finitely many finite groups G, such that per(G)\(\geq \epsilon \cdot | G|\).