Weakly totally permutable products and Fitting classes

Document Type : Research Paper

Author

Department of Mathematics and Applied Mathematics, University of Pretoria, Private bag X20, Hatfield, 0028, Pretoria, South Africa

Abstract

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.

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