A note on groups with a finite number of pairwise permutable seminormal subgroups

Document Type : Research Paper

Author

Department of Mathematics and Programming Technologies, Francisk Skorina Gomel State University, Gomel, Belarus

Abstract

A subgroup $A$ of a group $G$ is called {\it seminormal} in $G$‎, ‎if there exists a subgroup $B$ such that $G=AB$ and $AX$~is a subgroup of $G$ for every‎ ‎subgroup $X$ of $B$‎. ‎The group $G = G_1 G_2 \cdots G_n$ with pairwise permutable subgroups $G_1‎,‎\ldots‎,‎G_n$ such that $G_i$ and $G_j$ are seminormal in~$G_iG_j$ for any $i‎, ‎j\in \{1,\ldots‎,‎n\}$‎, ‎$i\neq j$‎, ‎is studied‎. ‎In particular‎, ‎we prove that if $G_i\in \frak F$ for all $i$‎, ‎then $G^\frak F\leq (G^\prime)^\frak N$‎, ‎where $\frak F$ is a saturated formation and $\frak U \subseteq \frak F$‎. ‎Here $\frak N$ and $\frak U$‎~ ‎are the formations of all nilpotent and supersoluble groups respectively‎, ‎the $\mathfrak F$-residual $G^\frak F$ of $G$ is the intersection of all those normal‎ ‎subgroups $N$ of $G$ for which $G/N \in \mathfrak F$‎.

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References

[1] B. Huppert, Endliche Gruppen I, Springer, Berlin, Heidelberg, New York, 1967.
[2] H. G. Bray and et al., Between Nilpotent and Soluble, Polygonal Publishing House, Passaic, 1982.
[3] K. Doerk and T. Hawkes, Finite soluble groups, Walter de Gruyter, Berlin, New York, 1992.
[4] A. Carocca, p-supersolvability of factorized finite groups, Hokkaido Math. J., 21 (1992) 395–403.
[5] A. Ballester-Bolinches, R. Esteban-Romero and M. Asaad, Products of finite groups, de Gruyter Expositions in
Mathematics, 53, Walter de Gruyter GmbH & Co. KG, Berlin, 2010.
[6] A. Ballester-Bolinches, J. C. Beidleman, H. Heineken and M. C. Pedraza-Aguilera, A survey on pairwise mutually
permutable products of finite groups, Algebra Discrete Math., 4 (2009) 1–9.
[7] A. Ballester-Bolinches, J. C. Beidleman, H. Heineken and M. C. Pedraza-Aguilera, On pairwise mutually permutable
products, Forum Math., 21 (2009) 1081–1090.
[8] A. Carocca, R. Maier, Theorems of Kegel-Wielandt type, Groups St. Andrews 1997 in Bath I, London Math. Soc.
Lecture Note Ser, 260, Cambridge University Press, Cambridge, 1999 195–201.
[9] X. Su, On semi-normal subgroups of finite group, J. Math. (Wuhan), 8 (1988) 7–9.
[10] A. Carocca and H. Matos, Some solvability criteria for finite groups, Hokkaido Math. J., 26 (1997) 157–161.
[11] V. S. Monakhov, Finite groups with a seminormal Hall subgroup, Math. Notes, 80 (2006) 542–549.
[12] V. N. Knyagina and V. S. Monakhov, Finite groups with seminormal Schmidt subgroups, Algebra and Logic, 46
(2007) 244–249.
[13] V. S. Monakhov and A. A. Trofimuk, Finite groups with two supersoluble subgroups, J. Group Theory, 22 (2019)
297–312.
[14] V. V. Podgornaya, Seminormal subgroups and supersolubility of finite groups, Vesti Akad. Navuk Belarusi Ser.
Fiz.-Mat. Navuk, 4 (2000) 22–25.
[15] W. Guo, Finite groups with seminormal Sylow subgroups, Acta Mathematica Sinica, 24 (2008) 1751–1758.
[16] V. S. Monakhov and A. A. Trofimuk, On the supersolubility of a group with seminormal subgroups, Siberian Math.
J., 61 (2020) 118–126.
[17] A. Ballester-Bolinches and L. M. Ezquerro, Classes of Finite Groups, Dordrecht, Springer, 2006.
[18] M. Asaad and A. Shaalan, On the supersolubility of finite groups, Arch. Math., 53 (1989) 318–326.
[19] V. S. Monakhov, On the supersoluble residual of mutually permutable products, PFMT, 34 (2018) 69–70.

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