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On the derived length of groups with some permutational property. (English) Zbl 0721.20022
A group G has property $$Q_ n$$ if for every choice $$x_ 1,...,x_ n\in G$$ there exist distinct permutations $$\sigma$$, $$\tau$$ in $$S_ n$$ such that $x_{\sigma (1)}...x_{\sigma (n)}=x_{\tau (1)}...x_{\tau (n)}.$ If one can always choose $$\tau =1$$, then G is said to have $$P_ n$$. It is shown that groups with $$Q_ 3$$ are metabelian. Moreover, the derived length of soluble groups with $$Q_ n$$ (resp. $$P_ n)$$ is $$\leq (n-2)^ 2(n+1)/2$$ (resp. $$\leq 2n-6)$$. The latter is one of the few results in this area that are valid for all n. It seems open whether the bounds given are best possible for $$n\geq 4$$ (resp. $$n\geq 5)$$.

##### MSC:
 20F16 Solvable groups, supersolvable groups 20F24 FC-groups and their generalizations 20E10 Quasivarieties and varieties of groups 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20F12 Commutator calculus 20F05 Generators, relations, and presentations of groups 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks
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