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A permutational property of groups. (English) Zbl 0544.20036
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.

MSC:
 20F24 FC-groups and their generalizations 20E07 Subgroup theorems; subgroup growth 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20F05 Generators, relations, and presentations of groups
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References:
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