The supersolvable residual of a finite group factorized by pairwise permutable seminormal subgroups

A. A. Trofimuk

Algebra i logika, Tome 60 (2021) no. 3, pp. 313-326

Résumé

A subgroup $A$ is seminormal in a finite group $G$ if there exists a subgroup $B$ such that $G=AB$ and $AX$ is a subgroup for each subgroup $X$ from $B$. We study a group $G=G_1G_2\ldots G_n$ with pairwise permutable supersolvable groups $G_1,\ldots,G_n$ such that $G_i$ and $G_j$ are seminormal in $G_iG_j$ for any $i,j\in\{1,\ldots,n\}$, $i\neq j$. It is stated that $G^\mathfrak U=(G^\prime)^\mathfrak N$. Here $\mathfrak N$ and $\mathfrak U$ are the formations of all nilpotent and supersolvable groups, and $H^\mathfrak X$ and $H^{\prime}$ are the $\mathfrak X$-residual and the derived subgroup, respectively, of a group $H$. It is proved that a group $G=G_1G_2\ldots G_n$ with pairwise permutable subgroups $G_1,\ldots,G_n$ is supersolvable provided that all Sylow subgroups of $G_i$ and $G_j$ are seminormal in $G_iG_j$ for any $i,j\in\{1,\ldots,n\}$, $i\neq j$.

Détail

Mots-clés : supersolvable group
Keywords: nilpotent group, seminormal subgroup, derived subgroup, $\mathfrak X$-residual, Sylow subgroup.

Bibliographie

[1] A. Ballester-Bolinches, R. Esteban-Romero, M. Asaad, Products of finite groups, De Gruyter Exp. Math., 53, Walter de Gruyter, Berlin, 2010 | DOI | Zbl

[2] B. Huppert, Endliche Gruppen, v. I, Grundlehren Math. Wiss., 134, Springer-Verlag, Berlin a.o., 1967 | DOI | Zbl

[3] O. H. Kegel, “Produkte nilpotenter Gruppen”, Arch. Math., 12 (1961), 90–93 | DOI | Zbl

[4] L. S. Kazarin, “Priznaki neprostoty faktorizuemykh grupp”, Izv. AN SSSR. Ser. matem., 44:2 (1980), 288–308 | Zbl

[5] B. Huppert, “Monomiale Darstellung endlicher Gruppen”, Nagoya Math. J., 6 (1953), 93–94 | DOI | Zbl

[6] R. Baer, “Classes of finite groups and their properties”, Ill. J. Math., 1 (1957), 115–187 | Zbl

[7] D. K. Friesen, “Products of normal supersolvable subgroups”, Proc. Am. Math. Soc., 30:1 (1971), 46–48 | DOI | Zbl

[8] A. F. Vasilev, T. I. Vasileva, “O konechnykh gruppakh, u kotorykh glavnye faktory yavlyayutsya prostymi gruppami”, Izv. vuzov. Matem., 1997, no. 11, 10–14

[9] V. S. Monakhov, I. K. Chirik, “O sverkhrazreshimom koradikale proizvedeniya subnormalnykh sverkhrazreshimykh podgrupp”, Sib. matem. zh., 58:2 (2017), 353–364 | Zbl

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

[11] L. S. Kazarin, “Faktorizatsii konechnykh grupp razreshimymi podgruppami”, Ukr. matem. zh., 43:7–8 (1991), 947–950

[12] B. Amberg, L. C. Kazarin, B. Khefling, “Konechnye gruppy s kratnymi faktorizatsiyami”, Fundament. i prikl. matem., 4:4 (1998), 1251–1263 | Zbl

[13] V. S. Monakhov, A. A. Trofimuk, “O sverkhrazreshimosti gruppy s polunormalnymi podgruppami”, Sib. matem. zh., 61:1 (2020), 148–159 | Zbl

[14] K. Doerk, T. Hawkes, Finite soluble groups, De Gruyter Exp. Math., 4, Walter de Gruyter, Berlin etc., 1992

[15] V. S. Monakhov, “Konechnye gruppy s polunormalnoi khollovoi podgruppoi”, Matem. zametki, 80:4 (2006), 573–581 | Zbl

[16] V. S. Monakhov, A. A. Trofimuk, “On finite factorized groups with permutable subgroups of factors”, Arch. Math., 116:3 (2021), 241–249 | DOI | Zbl

[17] H. Wielandt, “Subnormalität in faktorisierten endlichen Gruppen”, J. Algebra, 69 (1981), 305–311 | DOI | Zbl

[18] V. S. Monakhov, Vvedenie v teoriyu konechnykh grupp i ikh klassov, Vysheishaya shkola, Minsk, 2006