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GENERALISED MUTUALLY PERMUTABLE PRODUCTS AND SATURATED FORMATIONS, II
Published online by Cambridge University Press: 30 January 2024
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.
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- © The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
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.