Geodesic
The $D_\pi$-Property in a~Class of Finite Groups
Algebra i logika, Tome 41 (2002) no. 3, pp. 335-370.

Voir la notice de l'article provenant de la source Math-Net.Ru

A finite group $G$ is a $D_\pi$-group for some set $\pi$ of primes if maximal $\pi$-subgroups of $G$ are all conjugate. Assume that every non-Abelian composition factor of the $D_\pi$-group $G$ is isomorphic either to an alternating group, or to one of the sporadic groups, or to a simple group of Lie type over a field whose characteristic belongs to $\pi$. We prove that an extension of $G$ by an arbitrary $D_\pi$-group and every normal subgroup of $G$ are $D_\pi$-groups. This gives partial answers to Questions 3.62 and 13.33 in the “Kourovka Notebook”. Also, we describe all $D_\pi$-groups whose composition factors are isomorphic to alternating, sporadic, and Lie-type groups whose characteristics belong to $\pi$. And bring to a close the description of Hall subgroups in sporadic groups, initiated by F. Gross.
Mots-clés : $D_\pi$-group, sporadic group
Keywords: alternating group, simple group of Lie type, Hall subgroup. 
@article{AL_2002_41_3_a4,
     author = {D. O. Revin},
     title = {The $D_\pi${-Property} in {a~Class} of {Finite} {Groups}},
     journal = {Algebra i logika},
     pages = {335--370},
     publisher = {mathdoc},
     volume = {41},
     number = {3},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2002_41_3_a4/}
}
TY  - JOUR
AU  - D. O. Revin
TI  - The $D_\pi$-Property in a~Class of Finite Groups
JO  - Algebra i logika
PY  - 2002
SP  - 335
EP  - 370
VL  - 41
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AL_2002_41_3_a4/
LA  - ru
ID  - AL_2002_41_3_a4
ER  - 
%0 Journal Article
%A D. O. Revin
%T The $D_\pi$-Property in a~Class of Finite Groups
%J Algebra i logika
%D 2002
%P 335-370
%V 41
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AL_2002_41_3_a4/
%G ru
%F AL_2002_41_3_a4
D. O. Revin. The $D_\pi$-Property in a~Class of Finite Groups. Algebra i logika, Tome 41 (2002) no. 3, pp. 335-370. http://geodesic.mathdoc.fr/item/AL_2002_41_3_a4/

[1] P. Hall, “Theorems like Sylow's”, Proc. Lond. Math. Soc.(III), 6:22 (1956), 286–304 | DOI | MR | Zbl

[2] Kourovskaya tetrad. Nereshennye voprosy teorii grupp, 15-e izd., In-t matem. SO RAN, Novosibirsk, 2002 | MR

[3] L. A. Shemetkov, “Silovskie svoistva konechnykh grupp”, Matem. sb., 76(118):2 (1968), 271–287 | Zbl

[4] L. A. Shemetkov, “O silovskikh svoistvakh konechnykh grupp”, Doklady AN BSSR, 16:10 (1972), 881–883 | Zbl

[5] V. D. Mazurov, D. O. Revin, “O khollovom $D_\pi$-svoistve dlya konechnykh grupp”, Sib. matem. zh., 38:1 (1997), 125–134 | MR | Zbl

[6] F. Gross, “On the existence of Hall subgroup”, J. Algebra, 98:1 (1986), 1–13 | DOI | MR | Zbl

[7] F. Gross, “On a conjecture of Philip Hall”, Proc. Lond. Math. Soc. (III), 52:3 (1986), 464–494 | DOI | MR | Zbl

[8] F. Gross, “Conjugacy of odd order Hall subgroups”, Bull. Lond. Math. Soc., 19:4(79) (1987), 311–319 | DOI | MR | Zbl

[9] A. S. Kondratev, “Podgruppy konechnykh grupp Shevalle”, Uspekhi matem. n., 41:1 (1986), 57–96 | MR

[10] R. W. Carter, Simple groups of Lie type, Wiley, London, 1972 | MR | Zbl

[11] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of finite groups, Clarendon Press, Oxford, 1985 | MR | Zbl

[12] D. Gorenstein, Konechnye prostye gruppy. Vvedenie v ikh klassifikatsiyu, Mir, M., 1985 | MR | Zbl

[13] H. Wielandt, “Zum Satz von Sylow”, Math. Z., 60:4 (1954), 407–408 | DOI | MR | Zbl

[14] J. E. Humphreys, Conjugacy classes in semisimple algebraic groups, Math. Surv. Monogr., 43, Am. Math. Soc., Providence, RI, 1995 | MR | Zbl

[15] D. Gorenstein, R. Lyons, “The local structure of finite groups of characteristic $2$ type”, Mem. Am. Math. Soc., 42:276 (1983) | MR

[16] W. Feit, J. G. Thompson, “Solvability of groups of odd order”, Pac. J. Math., 13:3 (1963), 775–1029 | MR | Zbl

[17] F. Gross, “Hall subgroups of order not divisible by $3$”, Rocky Mt. J. Math., 23:2 (1993), 569–591 | DOI | MR | Zbl

[18] D. O. Revin, “Khollovy $\pi$-podgruppy konechnykh grupp Shevalle, kharakteristika kotorykh prinadlezhit $\pi$”, Matem. trudy, 2:1 (1999), 160–208 | MR | Zbl

[19] M. Aschbacher, “Overgroups of Sylow subgroups in sporadic groups”, Mem. Am. Math. Soc., 60:343 (1986) | MR

[20] J. G. Thompson, “Hall subgroups of the symmetric groups”, J. Comb. Theory, 1 (1966), 271–279 | DOI | MR | Zbl