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GENERALISED MUTUALLY PERMUTABLE PRODUCTS AND SATURATED FORMATIONS, II

Part of: Representation theory of groups

Published online by Cambridge University Press:  30 January 2024

ADOLFO BALLESTER-BOLINCHES Open the ORCID record for ADOLFO BALLESTER-BOLINCHES [Opens in a new window] ,
SESUAI Y. MADANHA Open the ORCID record for SESUAI Y. MADANHA [Opens in a new window] ,
TENDAI M. MUDZIIRI SHUMBA Open the ORCID record for TENDAI M. MUDZIIRI SHUMBA [Opens in a new window]  and
MARÍA C. PEDRAZA-AGUILERA Open the ORCID record for MARÍA C. PEDRAZA-AGUILERA [Opens in a new window]
ADOLFO BALLESTER-BOLINCHES
Affiliation:
Departament de Matemàtiques, Universitat de València, Dr. Moliner 50, 46100 Burjassot, València, Spain e-mail: adolfo.ballester@uv.es
SESUAI Y. MADANHA*
Affiliation:
Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa
TENDAI M. MUDZIIRI SHUMBA
Affiliation:
Sobolev Institute of Mathematics, Novosibirsk, Russia e-mail: tendshumba@gmail.com
MARÍA C. PEDRAZA-AGUILERA
Affiliation:
Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, 46022 Camino de Vera, València, Spain e-mail: mpedraza@mat.upv.es
*
e-mail: sesuai.madanha@up.ac.za
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Abstract

A group $G=AB$ is the weakly mutually permutable product of the subgroups A and B, if A permutes with every subgroup of B containing $A \cap B$ and B permutes with every subgroup of A containing $A \cap B$. Weakly mutually permutable products were introduced by the first, second and fourth authors [‘Generalised mutually permutable products and saturated formations’, J. Algebra 595 (2022), 434–443] who showed that if $G'$ is nilpotent, A permutes with every Sylow subgroup of B and B permutes with every Sylow subgroup of A, then $G^{\mathfrak {F}}=A^{\mathfrak {F}}B^{\mathfrak {F}} $, where $ \mathfrak {F} $ is a saturated formation containing $ \mathfrak {U} $, the class of supersoluble groups. In this article we prove results on weakly mutually permutable products concerning $ \mathfrak {F} $-residuals, $ \mathfrak {F} $-projectors and $\mathfrak {F}$-normalisers. As an application of some of our arguments, we unify some results on weakly mutually $sn$-products.

Keywords

weakly mutually permutable productssupersoluble groupssaturated formationsprojectorsnormalisers

MSC classification

Primary: 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks
Secondary: 20D20: Sylow subgroups, Sylow properties, $pi$-groups, $pi$-structure
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 11
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The work of the third author is supported by the Mathematical Center in Akademgorodok under agreement no. 075-15-2022-281 with the Ministry of Science and Higher Education of the Russian Federation.

References

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