×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] A. Ballester-Bolinches, R. Esteban-Romero and M. Asaad,Products of Finite Groups, Walter De Gruyter, BerlinNew York, 2010. · Zbl 1206.20019
[2] A. Ballester-Bolinches, M. C. Pedraza-Aguilera and M. D. P´erez-Ramos, Finite groups which are products of pairwise totally permutable subgroups,Proc. of the Edin. Math. Soc.,41(1998) 567-572. · Zbl 0904.20013
[3] J. C. Beidleman and H. Heineken, Mutually permutable subgroups and group classes,Arch. Math.,85(2005) 18-30. · Zbl 1103.20015
[4] J. Bochtler and P. Hauck, Mutually permutable subgroups and Fitting classes,Arch. Math.,88(2007) 385-388. · Zbl 1125.20009
[5] K. Doerk and T. O. Hawkes,Finite Soluble Groups, Walter De Gruyter, Berlin-New York, (1992). · Zbl 0753.20001
[6] P. Hauck, A. Mart´ınez-Pastor and M. D. P´erez-Ramos, Fitting classes and products of totally permutable groups, J. Algebra,252(2002) 114-126. · Zbl 1014.20010
[7] S.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.