V.S. Monakhov, V.N. Tyutyanov. Finite groups with supersoluble subgroups of given orders ... P. 155-163

We study a finite group $G$ with the following property: for any of its maximal subgroups $H$, there exists a subgroup $H_1$ such that $|H_1|=|H|$ and $H_1\in \frak F$, where $\frak F$ is the formation of all nilpotent groups or all supersoluble groups. We prove that, if $\frak F=\frak N$ is the formation of all nilpotent groups and $G$ is nonnilpotent, then $|\pi (G)|=2$ and $G$ has a normal Sylow subgroup. For the formation $\frak F=\frak U$ of all supersoluble groups and a soluble group $G$ with the above property, we prove that $G$ is supersoluble, or $2\le |\pi (G)|\le 3$; if $|\pi (G)|=3$, then $G$ has a Sylow tower of supersoluble type; if $|\pi (G)|=2$, then either $G$ has a normal Sylow subgroup or, for the largest $p\in \pi (G)$, some maximal subgroup of a Sylow $p$-subgroup is normal in $G$. If $G$ is nonsoluble and, for each maximal subgroup of $G$, there exists a supersoluble subgroup of the same order, then every nonabelian composition factor of $G$ is isomorphic to $PSL_2(p)$ for some prime $p$; we list all such values of $p$.

Keywords: finite group, soluble group, maximal subgroup, nilpotent subgroup, supersoluble subgroup

Received April 15, 2019

Revised June 27, 2019

Accepted July 8, 2019

Viktor Stepanovich Monakhov, Dr. Phys.-Math. Sci., Prof., Francisk Skorina Gomel State University, Gomel, 246019, Republic of Belarus, e-mail: victor.monakhov@gmail.com

Valentin Nikolayevich Tyutyanov, Dr. Phys.-Math. Sci., Prof., Gomel Branch of International University “MITSO”, Gomel, 246029, Republic of Belarus, e-mail: vtutanov@gmail.com

REFERENCES

1.   Schmidt O. Groups, all subgroups of which are special. Mat. Sb., 1924, vol. 31, no. 3–4, pp. 366–372 (in Russian).

2.   Doerk K. Minimal nicht $\ddot{\mathrm{u}}$beraufl$\ddot{\mathrm{o}}$sbare, endliche Gruppen. Math. Z., 1966, vol. 91, no. 3, pp. 198–205. doi: 10.1007/BF01312426 

3.   Thompson J.G. Nonsolvable finite groups all of whose local subgroups are solvable. Bull. Amer. Math. Soc., 1968, vol. 74, pp. 383–437. doi: 10.1090/S0002-9904-1968-11953-6 

4.   Shemetkov L.A. Formatsii konechnykh grupp [Formations of finite groups]. Minsk: Nauka Publ., 1978, 271 p.

5.   Monakhov V.S., Tyutyanov V.N. On finite groups with some subgroups of prime indices. Siberian Math. J., 2007, vol. 48, no. 4, pp. 666–668. doi: 10.1007/s11202-007-0068-3 

6.   Huppert B. Endliche Gruppen I. Berlin; Heidelberg; N Y: Springer, 1967, 796 p. doi: 10.1007/978-3-642-64981-3 .

7.   Vdovin E.P. Carter subgroups of finite groups. Siberian Adv. Math., 2009, vol. 19, no. 1, pp. 24–74. doi: 10.3103/S1055134409010039 

8.   Kazarin L.S., Korzyukov Yu.A. Finite solvable groups with supersolvable maximal subgroups. Soviet Mathematics (Izvestiya VUZ. Matematika), 1980, vol. 24, no. 5, pp. 23–29.

9.   Gorenstein D. Finite groups. N Y: Harper and Row, 1968. 519 p.

10.   Kondrat’ev A.S. Subgroups of finite Chevalley groups. Russian Math. Surveys, 1986, vol. 41, no. 1, pp. 65–118. doi: 10.1070/RM1986v041n01ABEH003203 

11.   Li S., Shi W. A note on the solvability of groups. J. Algebra, 2006, vol. 304, no. 1, pp. 278–285. doi: 10.1016/j.jalgebra.2005.09.028 

12.   Seitz G.M. Flag-transitive subgroups of Chevalley groups. Ann. Math., 1973, vol. 97, no. 1, pp. 27–56. doi: 10.2307/1970876 

13.   Gorenstein D., Lyons R. The local structure of the finite groups of characteristic 2 type. Mem. Amer. Math. Soc, vol. 42, 731 p. doi: 10.1090/memo/0276 

14.   Baumann B. Endliche nichtaufl$\ddot{\mathrm{o}}$sbare Gruppen mit einer nilpotenten maximalen Untergruppen. J. Algebra, 1975, vol. 38, no. 1, pp. 119–135. doi: 10.1016/0021-8693(76)90249-0 

15.   Thompson J.G. A special class of non solvable groups. Math. Z., 1960, vol. 72, no. 1, pp. 458–462. doi: 10.1007/BF01162968 

16.   Gorenstein D. Finite simple groups. An introduction to their classification. University Series in Mathematics, N Y: Plenum Publishing Corp., 1982, 333 p. ISBN: 0-306-40779-5 . Translated to Russian under the title Konechnye prostye gruppy. Vvedenie v ikh klassifikatsiyu. Moscow: Mir Publ., 1985, 352 p.

17.   Mazurov V.D. The minimal permutation representation of the Thompson simple group. Algebra and Logic, 1988, vol. 27, no 5, pp. 350–361. doi: 10.1007/BF01982274 

18.   Hall P. Theorems like Sylow‘s. Proc. London Math. Soc., 1956, vol. s3-6, no. 2, pp. 286–304. doi: 10.1112/plms/s3-6.2.286 

19.   Guralnick R.M. Subgroups of prime power index in a simple group. J. Algebra, 1983, vol. 81, no. 2, pp. 304–311. doi: 10.1016/0021-8693(83)90190-4 

Cite this article as: V.S.Monakhov, V.N.Tyutyanov. Finite groups with supersoluble subgroups of given orders, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 4, pp. 155–163.