Abstract
We say a group G = AB is the totally semipermutable product of subgroups A and B if every Sylow subgroup P of A is totally permutable with every Sylow subgroup Q of B whenever \( \gcd(|P|,|Q|)=1 \). Products of pairwise totally semipermutable subgroups are studied in this article. Let \( \mathfrak{U} \) denote the class of supersoluble groups and \( \mathfrak{D} \) denote the formation of all groups which have an ordered Sylow tower of supersoluble type. We obtain the \( \mathfrak{F} \)-residual of the product from the \( \mathfrak{F} \)-residuals of the pairwise totally semipermutable subgroups when \( \mathfrak{F} \) is a subgroup-closed saturated formation such that \( \mathfrak{U}\subseteq \mathfrak{F}\subseteq \mathfrak{D} \).
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References
M. Asaad, On a new type of products of finite groups, Publ. Math. Debrecen, 97 (2020), 497–505.
M. Asaad and A. Shaalan, On the supersolvability of finite groups, Arch. Math., 53 (1989), 318–326.
A. Ballester-Bolinches, R. Esteban-Romero and M. Asaad Products of Finite Groups, Walter De Gruyter (Berlin–New York, 2010).
A. Ballester-Bolinches, M. C. Pedraza-Aguilera and M. D. Pérez-Ramos, Finite groups which are products of pairwise totally permutable subgroups, Proc. Edinburgh Math. Soc., 41 (1998), 567–572.
A. Ballester-Bolinches and M. D. Pérez-Ramos, A question of R. Maier concerning formations, J. Algebra, 182 (1996), 738—747.
K. Doerk and T. O. Hawkes, Finite Soluble Groups, Walter De Gruyter (Berlin–New York, 1992).
B. Huppert, Endliche Gruppen. I, Springer Verlag (Berlin–New York, 1967).
V. S. Monakhov and A. A. Trofimuk, On finite factorized groups with permutable subgroups of factors, Arch. Math., 116 (2021), 241–249.
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The third author was supported by the National Research Foundation (NRF) of South Africa under Grant Number 150857. All opinions, findings and conclusions or recommendations expressed in this publication are those of the authors and the NRF does not accept any liability in regard thereto.
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Ballester-Bolinches, A., Cossey, J., Madanha , S.Y. et al. On totally semipermutable products of finite groups. Acta Math. Hungar. 172, 161–169 (2024). https://doi.org/10.1007/s10474-024-01392-4
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DOI: https://doi.org/10.1007/s10474-024-01392-4