Comptes Rendus
Algebraic geometry
On homogeneous spaces with finite anti-solvable stabilizers
Giancarlo Lucchini Arteche1
1 Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Las Palmeras 3425, Ñuñoa, Santiago, Chile
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 777-780

We say that a group is anti-solvable if all of its composition factors are non-abelian. We consider a particular family of anti-solvable finite groups containing the simple alternating groups for n≠6 and all 26 sporadic simple groups. We prove that, if K is a perfect field and X is a homogeneous space of a smooth algebraic K-group G with finite geometric stabilizers lying in this family, then X is dominated by a G-torsor. In particular, if G=SL n , all such homogeneous spaces have rational points.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.339
Classification: 14M17, 12G05, 14G05
Keywords: homogeneous spaces, rational points, non-abelian cohomology, finite simple groups

Giancarlo Lucchini Arteche 1

1 Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Las Palmeras 3425, Ñuñoa, Santiago, Chile
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{CRMATH_2022__360_G7_777_0,
     author = {Giancarlo Lucchini Arteche},
     title = {On homogeneous spaces with finite anti-solvable stabilizers},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {777--780},
     year = {2022},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     doi = {10.5802/crmath.339},
     language = {en},
}
TY  - JOUR
AU  - Giancarlo Lucchini Arteche
TI  - On homogeneous spaces with finite anti-solvable stabilizers
JO  - Comptes Rendus. Mathématique
PY  - 2022
SP  - 777
EP  - 780
VL  - 360
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.339
LA  - en
ID  - CRMATH_2022__360_G7_777_0
ER  - 
%0 Journal Article
%A Giancarlo Lucchini Arteche
%T On homogeneous spaces with finite anti-solvable stabilizers
%J Comptes Rendus. Mathématique
%D 2022
%P 777-780
%V 360
%I Académie des sciences, Paris
%R 10.5802/crmath.339
%G en
%F CRMATH_2022__360_G7_777_0
Giancarlo Lucchini Arteche. On homogeneous spaces with finite anti-solvable stabilizers. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 777-780. doi: 10.5802/crmath.339

[1] Ronald D. Bercov On groups without abelian composition factors, J. Algebra, Volume 5 (1967), pp. 106-109 | MR | Zbl | DOI

[2] Mikhail V. Borovoi Abelianization of the second nonabelian Galois cohomology, Duke Math. J., Volume 72 (1993) no. 1, pp. 217-239 | Zbl | MR

[3] Cyril Demarche; Giancarlo Lucchini Arteche Le principe de Hasse pour les espaces homogènes : réduction au cas des stabilisateurs finis, Compos. Math., Volume 155 (1900) no. 8, pp. 1568-1593 | Zbl | DOI

[4] David S. Dummit; Richard M. Foote Abstract Algebra, John Wiley & Sons, 2004

[5] Yuval Z. Flicker; Claus Scheiderer; R. Sujatha Grothendieck’s theorem on non-abelian H 2 and local-global principles, J. Am. Math. Soc., Volume 11 (1998) no. 3, pp. 731-750 | Zbl | MR | DOI

[6] Daniel Gorenstein Finite Groups, Chelsea Publishing, 1980

[7] Andrea Lucchini; Federico Menegazzo; Marta Morigi On the existence of a complement for a finite simple group in its automorphism group, Ill. J. Math., Volume 47 (2003) no. 1-2, pp. 395-418 | MR | DOI

[8] Saunders Mac Lane Homology, Classics in Mathematics, Springer, 1995 (Reprint of the 1975 edition)

[9] Tonny A. Springer Nonabelian H 2 in Galois cohomology, Algebraic Groups and Discontinuous Subgroups (Proceedings of Symposia in Pure Mathematics), Volume 9, American Mathematical Society, 1966, pp. 164-182 | Zbl | DOI

[10] Karl-Heinz Ulbrich On nonabelian H 2 for profinite groups, Can. J. Math., Volume 43 (1991) no. 1, pp. 213-224 | Zbl | MR | DOI

Cited by Sources:

Comments - Policy