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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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