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On the 4-permutational property. (English) Zbl 0623.20022
The finite group lies in the class $$P_ 4$$ if, given elements $$x_ 1,...,x_ 4$$ of G, there is a permutation $$\sigma$$ of $$\{$$ 1,2,3,4$$\}$$ such that $$x_{1\sigma}x_{2\sigma}x_{3\sigma}x_{4\sigma}=x_ 1x_ 2x_ 3x_ 4$$. Using the N-group paper, it is shown that all such groups are soluble. Further, it is shown by less regal means that for such a group G, the exponent of the Sylow 2-subgroups of G/F(G) is a divisor of 4, G’ is nilpotent, and G/F(G) is an Abelian $$\{$$ 2,3$$\}$$- group. No motivation for or application of these results is given.
Reviewer: N.Blackburn

MSC:
 20F05 Generators, relations, and presentations of groups 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20F12 Commutator calculus 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 20D15 Finite nilpotent groups, $$p$$-groups 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure 20F24 FC-groups and their generalizations
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