A Schmidt group is a finite nonnilpotent group each of whose proper subgroups is nilpotent. A supplement of a subgroup $A$ in a group $G$ is a subgroup $B$ of $G$ such that $G=AB$. Finite groups in which a Sylow subgroup is permutable with some Schmidt subgroups were studied by Ya.G. Berkovich and E.M. Pal'chik (Sib. Mat. Zh. 8(4), 741-753 (1967)) and by V. N. Knyagina and V.S. Monakhov (Proc. Steklov Inst. Math. 272 (Suppl. 1), S55-S64 (2011)). In this situation, the group may be nonsolvable. For example, in the group PSL(2,7) a Sylow 2-subgroup is permutable with all Shmidt subgroups of odd order. In the group SL(2,8) a Sylow 3-subgroup is permutable with all 2-closed Shmidt subgroups of even order. In the group SL(2,4) a Sylow 5-subgroup is permutable with every 2-closed Shmidt subgroup of even order. Since the groups Sz$(2^{2k+1})$ for $k\geq 1$, PSU(5,4), PSU(4,2), and PSp$(4,2^n)$ do not contain Shmidt subgroups of odd order, in these groups any Sylow subgroup is permutable with any Shmidt subgroup of odd order. We establish the $r$-solvability a finite group $G$ such that $r$ is odd and is not a Fermat prime and a Sylow $r$-subgroup $R$ is permutable with 2-nilpotent (or 2-closed) Schmidt subgroups of even order from some supplement of $R$ in $G$. We give examples showing that the constraints on $r$ are not superfluous.
Keywords: finite group, Schmidt group, $r$-solvable group, Sylow $r$-subgroup.
The paper was received by the Editorial Office on April 27, 2018.
Viktor Stepanovich Monakhov, Dr. Phys.-Math. Sci., Prof., Francisk Skorina Gomel State University, Gomel, 246019, Republic of Belarus, e-mail: victor.monakhov@gmail.com
Ekaterina Vladimirovna Zubei, doctoral student, Francisk Skorina Gomel State University, Gomel, 246019, Republic of Belarus, e-mail: ekaterina.zubey@yandex.ru
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. 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).
3. 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 .
4. 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 .
5. Huppert B. Endliche Gruppen I. Berlin; Heidelberg; N Y: Springer, 1967, 793 p. doi: 10.1007/978-3-642-64981-3 .
6. Monakhov V.S. Vvedenie v teoriyu konechnykh grupp i ikh klassov [Introduction to the theory of finite groups and their classes]. Minsk: Vysheishaya Shkola, 2006, 207 p. (in Russian). ISBN: 985-06-1114-6 .
7. 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 .
8. Su X. On seminormal subgroups of finite group. J. Math. (Wuhan), 1988, vol. 8, no. 1, pp. 7–9.
9. Carocca A., Matos H. Some solvability criteria for finite groups. Hokkaido Math. J., 1997, vol. 26, no. 1, pp. 157–161. doi: 10.14492/hokmj/1351257811 .
10. Podgornaya V.V. Seminormal subgroups and supersolvability of finite groups. Vestsi Nats. Akad. Navuk Belarusi Ser. Fiz.-Mat. Navuk, 2000, no. 4, pp. 22–25.
11. Monakhov V.S. Finite groups with a seminormal Hall subgroup. Math. Notes., 2006, vol. 80, no. 4, pp. 542–549. doi: 10.4213/mzm2850 .
12. Berkovich Ya.G. A theorem on non-nilpotent solvable subgroups of a finite group. In: Konechnye gruppy, Berkovich Ya.G. (ed.), Minsk, Nauka i tekhnika Publ., 1966, 195 p., ISBN: 978-5-458-54866-3 , pp. 24–39 (in Russian).
13. Monakhov V.S. The Schmidt subgroups of finite groups. Voprosy Algebry, 1998, no. 13, pp. 153–171 (in Russian).
14. Ito N. On the faktorisations of the linear fractional group $LF(2,p^n)$. Acta Sci. Math., 1953, no. 15, pp. 79–84.
15. Monakhov V.S. The product of finite groups close to nilpotent. In: Finite Groups, Shemetkov L.A. (ed.). Minsk: Nauka i Tekhnika Publ., 1975, pp. 70–100 (in Russian).
16. Fisman E. On the product of two finite solvable groups. J. Algebra, 1983, vol. 80, pp. 517–536.
17. Kazarin L.S. On groups which are the product of two solvable subgroups. Commun. Algebra, 1986, vol. 14, no. 6, pp. 1001–1066. doi: 10.1080/00927878608823352 .
18. Conway J.H., Curtis R.T., Norton S.P., Parker R.A., Wilson R.A.. Atlas of finite groups, Oxford: Clarendon Press, 1985, 252 p. ISBN-10: 0198531990 .
19. Knyagina V.N., Monakhov V.S. On the π′-properties of a finite group possessing a hall π-subgroup. Sib. Math. J., 2011, vol. 52, no. 2, pp. 234–243. doi: 10.1134/S0037446611020066 .
