Оn the soluble radical of the finite groups

  • S. Yu. Bashun Polotz. state University, Belarus
  • E. M. Palchik Polotz. state University, Belarus

Abstract

UDC 512.542<br>

We assume that $G$ is a finite group, $\pi(G)=\{s\}\cup \sigma$, $s > 2$, $\Sigma$ is a set of Sylow$\sigma$-subgroups taken one for each $p_i\in \sigma$, $R(G)$ is the largest normal soluble subgroup in $G$ (the soluble radical of $G$). Suppose also that each Sylow $p_i$-subgroup $G_{p_i}\in \Sigma$ normalizes thes-subgroup $T^{(i)}\neq 1$ of the group $G$. With these assumptions, we determine the conditions under whichs divides $|R(G)|$.

 

References

V. N. Tyutyanov, L. A. Shemetkov, Тройные факторизации в конечных группах (Russian), Trojny`e faktorizaczii v konechny`kh gruppakh, Dokl. NAN Belarusi,46, No 4, 52 – 55 (2002).

E`.M. Pal`chik, О свойствах некоторых простых делителей порядков минизотропных торов конечных групп лиева типа (Russian) O svojstvakh nekotory`kh prosty`kh delitelej poryadkov minizotropny`kh torov konechny`kh grupp lieva tipa, Vesczi NAN Belarusi, ser. fiz.-mat. navuk, No 4, 66 – 71 (2012).

E`.M. Pal`chik, Конечные простые группы с факторизацией $ G = G_{ pi} B, 2 not inpi$ (Russian) Konechny`e prosty`e gruppy` s faktorizacziej $ G = G_{ pi} B, 2 not inpi$, Tr. In-ta matematiki i mekhaniki UrO RAN,20, No 2, 242 – 249 (2014).

Huppert, B. Endliche Gruppen. I. (German) Die Grundlehren der Mathematischen Wissenschaften, Band 134 Springer-Verlag, Berlin-New York 1967 {rm xii}+793 pp. MR0224703

D. Gorenstejn, Конечные простые группы. Введение в их классификацию (Russian) Konechny`e prosty`e gruppy`. Vvedenie v ikh klassifikacziyu, Mir, Moskva (1985)

Gorenstein, Daniel; Lyons, Richard. The local structure of finite groups of characteristic $2$ type. Mem. Amer. Math. Soc. 42 (1983), no. 276, {rm vii}+731 pp. doi: 10.1090/memo/0276

Gorenstein, Daniel; Lyons, Richard; Solomon, Ronald. The classification of the finite simple groups. Mathematical Surveys and Monographs, 40.1. American Mathematical Society, Providence, RI, 1994. {rm xiv}+165 pp. ISBN: 0-8218-0334-4 doi: 10.1090/surv/040.1

Wilson, Robert A. The finite simple groups. Graduate Texts in Mathematics, 251. Springer-Verlag London, Ltd., London, 2009. xvi+298 pp. ISBN: 978-1-84800-987-5 doi: 10.1007/978-1-84800-988-2

Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A. Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray. Oxford University Press, Eynsham, 1985. {rm xxxiv}+252 pp. ISBN: 0-19-853199-0 MR0827219

A. S. Kondrat`ev, V. D. Mazurov, 2-Сигнализаторы конечных простых групп (Russian) 2-Signalizatory` konechny`kh prosty`kh grupp, Algebra i logika,42, No 5 (2003), 594 – 623.

Berkovic, Ja. G. On $p$-subgroups of finite symmetric and alternating groups. Representation theory, group rings, and coding theory, 67--76, Contemp. Math., 93, Amer. Math. Soc., Providence, RI, 1989. doi: 10.1090/conm/093/1003342

Glauberman, G. Factorizations in local subgroups of finite groups. Regional Conference Series in Mathematics, No. 33. American Mathematical Society, Providence, R.I., 1977. {rm ix}+74 pp. ISBN: 0-8218-1683-7 MR0470072

Huppert, Bertram; Blackburn, Norman. Finite groups. III. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 243. Springer-Verlag, Berlin-New York, 1982. {rm ix}+454 pp. ISBN: 3-540-10633-2 MR0662826

E. P. Vdovin, D. O. Revin, Теоремы силовского типа (Russian)Teoremy` silovskogo tipa,Uspekhi mat. nauk,66, No 5(401), 3 – 46 (2011).

Arad, Zvi; Fisman, Elsa. On finite factorizable groups. J. Algebra 86 (1984), no. 2, 522--548. doi: 10.1016/0021-8693(84)90046-2

Li, Cai Heng; Li, Xianhua. On permutation groups of degree a product of two prime-powers. Comm. Algebra 42 (2014), no. 11, 4722--4743. doi: 10.1080/00927872.2013.823500

Lennox, John C.; Stonehewer, Stewart E. Subnormal subgroups of groups. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1987. {rm ix}+253 pp. ISBN: 0-19-853552-X MR0902587

Baer, Reinhold. Kriterien für die Zugehörigkeit von Elementen zu $Osb{omega }G$. Math. Z. 152 (1977), no. 3, 207--222. doi: 10.1007/BF01488965

S. Tchounikhin, Symplicite du groupe finiles orders de ces classes d’elements conjgues, C. r. Acad. Sci.,191, 397 – 399 (1930).

L. S. Kazarin, О проблеме С. А. Чунихина (Russian) Issledovaniya po teorii grupp, UNCz AN SSSR, Sverdlovsk (1984), s. 81 – 99

Published
28.03.2020
How to Cite
Bashun, S. Y., and E. M. Palchik. “Оn the Soluble Radical of the Finite Groups”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 3, Mar. 2020, pp. 326-39, doi:10.37863/umzh.v72i3.800.
Section
Research articles