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Weakly totally permutable products and Fitting classes. (English) Zbl 07324035
Summary: It is known that if \(G=AB\) is a product of its totally permutable subgroups \(A\) and \(B \), then \(G\in \mathfrak{F}\) if and only if \(A\in \mathfrak{F}\) and \(B\in \mathfrak{F}\) when \(\mathfrak{F}\) is a Fischer class containing the class \(\mathfrak{U}\) of supersoluble groups. We show that this holds when \(G=AB\) is a weakly totally permutable product for a particular Fischer class, \(\mathfrak{F}\diamond\mathfrak{N}\), where \(\mathfrak{F}\) is a Fitting class containing the class \(\mathfrak{U}\) and \(\mathfrak{N}\) a class of nilpotent groups. We also extend some results concerning the \(\mathfrak{U}\)-hypercentre of a totally permutable product to a weakly totally permutable product.
20D40 Products of subgroups of abstract finite groups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
Full Text: DOI
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