Geodesic
On a product of two formational $\mathrm{tcc}$-subgroups
Algebra and discrete mathematics, Tome 30 (2020) no. 2, pp. 282-289.

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A subgroup $A$ of a group $G$ is called $\mathrm{tcc}$-subgroup in $G$, if there is a subgroup $T$ of $G$ such that $G=AT$ and for any $X\le A$ and $Y\le T$ there exists an element $u\in \langle X,Y\rangle $ such that $XY^u\leq G$. The notation $H\le G $ means that $H$ is a subgroup of a group $G$. In this paper we consider a group $G=AB$ such that $A$ and $B$ are $\mathrm{tcc}$-subgroups in $G$. We prove that $G$ belongs to $\frak F$, when $A$ and $B$ belong to $\mathfrak F$ and $\mathfrak F$ is a saturated formation of soluble groups such that $\mathfrak U \subseteq \mathfrak F$. Here $\mathfrak U$ is the formation of all supersoluble groups.
Keywords: supersoluble group, totally permutable product, saturated formation, $\mathrm{tcc}$-permutable product, $\mathrm{tcc}$-subgroup. 
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A. Trofimuk. On a product of two formational $\mathrm{tcc}$-subgroups. Algebra and discrete mathematics, Tome 30 (2020) no. 2, pp. 282-289. http://geodesic.mathdoc.fr/item/ADM_2020_30_2_a11/

[1] B. Huppert, Endliche Gruppen, v. I, Springer, Berlin–Heidelberg–New York, 1967 | MR | Zbl

[2] V. S. Monakhov, Introduction to the Theory of Final Groups and Their Classes, Vysh. Shkola, Minsk, 2006 (Russian)

[3] K. Doerk and T. Hawkes, Finite soluble groups, Walter de Gruyter, Berlin–New York, 1992 | MR

[4] M. Asaad, A. Shaalan, “On the supersolubility of finite groups”, Arch. Math., 53 (1989), 318–326 | DOI | MR | Zbl

[5] R. Maier, “A completeness property of certain formations”, Bull. Lond. Math. Soc., 24 (1992), 540–544 | DOI | MR | Zbl

[6] A. Ballester-Bolinches, M. D. Perez-Ramos, “A question of R. Maier concerning formations”, J. Algebra, 182 (1996), 738–747 | DOI | MR | Zbl

[7] A. Carocca, “A note on the product of F-subgroups in a finite group”, Proc. Edinb. Math. Soc., 39 (1996), 37–42 | DOI | MR | Zbl

[8] A. Ballester-Bolinches, M. C. Pedraza-Aguilera, M. D. Pérez-Ramos, “Finite groups which are products of pairwise totally permutable subgroups”, Proc. Edinb. Math. Soc., 41 (1998), 567–572 | DOI | MR | Zbl

[9] W. Guo, K. P. Shum, A. N. Skiba, “Criterions of supersolubility for products of supersoluble groups”, Publ. Math. Debrecen, 68:3–4 (2006), 433–449 | MR | Zbl

[10] M. Arroyo-Jorda, P. Arroyo-Jorda, A. Martinez-Pastor, M. D. Perez-Ramos, “On finite products of groups and supersolubility”, J. Algebra, 323 (2010), 2922–2934 | DOI | MR | Zbl

[11] M. Arroyo-Jorda, P. Arroyo-Jorda, M. D. Perez-Ramos, “On conditional permutability and saturated formations”, Proc. Edinb. Math. Soc., 54 (2011), 309–319 | DOI | MR | Zbl

[12] M. Arroyo-Jorda, P. Arroyo-Jorda, A. Martinez-Pastor and M. D. Perez-Ramos, “On conditional permutability and factorized groups”, Annali di Matematica Pura ed Applicata, 193 (2014), 1123–1138 | DOI | MR | Zbl

[13] M. Arroyo-Jorda, P. Arroyo-Jorda, “Conditional permutability of subgroups and certain classes of groups”, Journal of Algebra, 476 (2017), 395–414 | DOI | MR | Zbl

[14] GAP, Groups, Algorithms, and Programming, Version 4.10.1, 2019 www.gap-system.org

[15] A. N. Skiba, “On weakly s-permutable subgroups of finite groups”, J. Algebra, 315 (2007), 192–209 | DOI | MR | Zbl