E.V. Zubei. On the solvability of a finite group with seminormal or subnormal Schmidt subgroups of one of its maximal subgroups ... P. 55-61

Vol. 25, no. 1, 2019

A Schmidt group is a finite non-nilpotent group all of whose proper subgroups are nilpotent. A group with a nilpotent maximal subgroup is known to be solvable if the derived subgroup of a Sylow 2-subgroup of a maximal subgroup is contained in the center of the Sylow 2-subgroup. If a maximal subgroup of a group is non-nilpotent, then it has a Schmidt subgroup. The structure of a group and, in particular, its solvability, depend on the properties of Schmidt subgroups of its maximal subgroup. In this paper, we establish the solvability of a finite group such that some Schmidt subgroups of its maximal subgroup are seminormal or subnormal in the group.

Keywords: finite group, solvable group, Schmidt subgroup, subnormal subgroup, seminormal subgroup, maximal subgroup

Received December 12, 2018

Revised January 30, 2019

Accepted February 4, 2019

Ekaterina Vladimirovna Zubei, doctoral student, Francisk Skorina Gomel State University, Gomel, 246019, Republic of Belarus, e-mail: ekaterina.zubey@yandex.ru

Cite this article as:   E.V. Zubei. On the solvability of a finite group with seminormal or subnormal Schmidt subgroups of one of its maximal subgroups , Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 1, pp. 55–61. 

REFERENCES

1.   Monakhov V.S. Vvedenie v teoriyu konechnykh grupp i ikh klassov [Introduction to the theory of finite groups and their classes]. Minsk: Vysheishaya Shkola Publ., 2006, 207 p. ISBN: 985-06-1114-6 .

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

3.   Thompson J. Finite groups with fixed point-free automorphisms of prime order. Proc. Nat. Sci., U.S.A., 1959, vol. 45, no. 4, pp. 578–581. doi: 10.1073/pnas.45.4.578 

4.   Thompson J. A special class of non-solvable groups. Math. Z., 1960, vol. 72, pp. 458–462. doi: 10.1007/BF01162968 

5.   Belonogov V.A. Solvability criterion for groups of even order. Sib. Math. J., 1966, vol. 7, no. 2, pp. 458–459 (in Russian).

6.   Monakhov V.S. Some solvability criteria of groups. DAN BSSR, 1970, vol. 14, no. 11, pp. 986–988 (in Russian).

7.   Monakhov V.S. Influence of properties of maximal subgroups on the structure of a finite group. Math. Notes, 1972, vol. 11, no. 2, pp. 115–118. doi: 10.1007/BF01097928 

8.   Baumann B. Endliche nichtaufl$\ddot{\mathrm{o}}$sbare gruppen mit einer nilpotenten maximal untergruppen. J. Algebra, 1976, vol. 38, pp. 119–135. doi: 10.1016/0021-8693(76)90249-0 

9.   Monakhov V.S. Finite groups with a seminormal Hall subgroup. Math. Notes, 2006, vol. 80, no. 4, pp. 542–549. doi: 10.4213/mzm2850 

10.   Guo W. Finite groups with seminormal Sylow subgroups. Acta Mathematica Sinica, 2008, vol. 24, no. 10, pp. 1751–1758. doi: 10.1007/s10114-008-6563-z 

11.   Knyagina V.N., Monakhov V.S. Finite groups with seminormal Schmidt subgroups. Algebra and Logic, 2007, vol. 46, no. 4, pp. 244–249. doi: 10.1007/s10469-007-0023-1 

12.   Knyagina V.N. On permutability of $n$ maximal subgroups with $p$-nilpotent Schmidt subgroups. Trudy Instituta Matematiki NAN Respubliki Belarus, 2016, vol. 24, no. 1, pp. 34–37 (in Russian).

13.   Berkovich Ya.G., Pal’chik, E.M. On the commutability of subgroups of a finite group. Sib. Math. J., 1967, vol. 8, no. 4, pp. 560–568. doi: 10.1007/BF02196475 

14.   Knyagina V.N., Monakhov V.S. On the permutability of Sylow subgroups with Schmidt subgroups. Proc. Steklov Institute Math., 2011, vol. 272, suppl. 1, pp. 55–64. doi: 10.1134/S0081543811020052 

15.   Zubei E.V., Knyagina V.N., Monakhov V.S. On the solvability of a finite group with $S$-seminormal Schmidt subgroups. Ukr. Mat. Zhurn., 2018, vol. 70, no. 11, pp. 1511–1518 (in Russian).

16.   Knyagina V.N., Monakhov V.S. Finite groups with subnormal schmidt subgroups. Siberian Math. J., 2004, vol. 45, no. 6, pp. 1075–1079. doi: 10.1023/B:SIMJ.0000048922.59466.20 

17.   Vedernikov V.A. Finite groups with subnormal Schmidt subgroups. Algebra and Logic, 2007, vol. 46, no. 6, pp. 363–372. doi: 10.1007/S10469-007-0036-9 

18.   Al-Sharo Kh. A., Skiba A.N. On finite groups with $\sigma$-subnormal Schmidt subgroups. Commun. Algebra, 2017, vol. 45, no. 10, pp. 4158–4165. doi: 10.1080/00927872.2016.1236938 

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

20.   Monakhov V.S. The Schmidt subgroups, their existence and some applications. Proc. Ukr. Math. Congr., 2001, Kiev, 2002, sec. 1, pp. 81–90 (in Russian).

21.   Berkovich Ya.G. A theorem on non-nilpotent solvable subgroups of a finite group. In: Finite groups, Berkovich Ya.G. (ed.), Minsk, Nauka i Tekhnika Publ., 1966, pp. 24–39 (in Russian). ISBN: 978-5-458-54866-3 .

22.   Monakhov V.S. The Schmidt subgroups of finite groups. Voprosy Algebry, 1998, no. 13, pp. 153–171 (in Russian).

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