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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.
MSC:
 20D40 Products of subgroups of abstract finite groups 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks
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References:
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