Conjugacy classes of finite subgroups of SL(2,F), SL(3,F ¯)
Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 3, pp. 555-571.

Let F is a field. We determine the finite subgroups G of SL(2,F) of cardinality |G| prime to the characteristic of F, up to conjugacy. When F=F ¯ is separably closed, using representation theory of finite groups we show that isomorphic subgroups of SL(2,F) are conjugate. We show this also for irreducible finite subgroups of SL(3,F ¯). The extension of the separably closed to the rational case is naturally based on Galois cohomology: we compute the first Galois cohomology group of the centralizer C of G in the SL, modulo the action of the normalizer. The results we obtain here in the semisimple simply connected case are different than those already known in the case of the adjoint group PGL(2). Finally, we determine the field of definition of such a finite subgroup G of SL(2,F ¯), that is, the minimal field F 1 with F 1 ¯=F ¯ such that the finite group G embeds in SL(2,F 1 ).

Soit F un corps. Nous déterminons les sous-groupes finis G de SL(2,F) dont le cardinal |G| n’est pas divisible par la caractéristique de F, à conjugaison près. Dans le cas où F=F ¯ est séparablement clos, nous montrons (via des arguments de la théorie des représentations des groupes finis) que deux sous-groupes isomorphes de SL(2,F) sont conjugués. Nous obtenons le même résultat pour les sous-groupes finis irréductibles de SL(3,F ¯). L’extension du cas séparablement clos au cas rationnel repose naturellement sur la cohomologie galoisienne. Plus précisément, nous calculons le premier groupe de cohomologie galoisienne du centralisateur C de G dans le SL en question, modulo l’action du normalisateur. Les résultats obtenus ici dans le cas semisimple simplement connexe sont différents des résultats déjà connus dans le cas du groupe adjoint PGL(2). Enfin, nous déterminons le corps de définition d’un tel sous-groupe fini G de SL(2,F ¯), c’est-à-dire le corps minimal F 1 , tel que F 1 ¯=F ¯ et tel que le groupe fini G s’injecte dans SL(2,F 1 ).

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1094
Classification: 14G05,  14G27,  14G25,  14E08
Keywords: Galois cohomology, SL(2,F), SL(3,F), algebraic classification, rational classification, representation theory
Flicker, Yuval Z. 1

1 Ariel University, Ariel 40700, Israel The Ohio State University, Columbus OH43210, USA
@article{JTNB_2019__31_3_555_0,
     author = {Flicker, Yuval Z.},
     title = {Conjugacy classes of finite subgroups of $\protect \mathrm{SL}(2,F)$, $\protect \mathrm{SL}(3,\protect \bar{F})$},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {555--571},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {3},
     year = {2019},
     doi = {10.5802/jtnb.1094},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.1094/}
}
TY  - JOUR
AU  - Flicker, Yuval Z.
TI  - Conjugacy classes of finite subgroups of $\protect \mathrm{SL}(2,F)$, $\protect \mathrm{SL}(3,\protect \bar{F})$
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2019
DA  - 2019///
SP  - 555
EP  - 571
VL  - 31
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - http://www.numdam.org/articles/10.5802/jtnb.1094/
UR  - https://doi.org/10.5802/jtnb.1094
DO  - 10.5802/jtnb.1094
LA  - en
ID  - JTNB_2019__31_3_555_0
ER  - 
%0 Journal Article
%A Flicker, Yuval Z.
%T Conjugacy classes of finite subgroups of $\protect \mathrm{SL}(2,F)$, $\protect \mathrm{SL}(3,\protect \bar{F})$
%J Journal de théorie des nombres de Bordeaux
%D 2019
%P 555-571
%V 31
%N 3
%I Société Arithmétique de Bordeaux
%U https://doi.org/10.5802/jtnb.1094
%R 10.5802/jtnb.1094
%G en
%F JTNB_2019__31_3_555_0
Flicker, Yuval Z. Conjugacy classes of finite subgroups of $\protect \mathrm{SL}(2,F)$, $\protect \mathrm{SL}(3,\protect \bar{F})$. Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 3, pp. 555-571. doi : 10.5802/jtnb.1094. http://www.numdam.org/articles/10.5802/jtnb.1094/

[1] Beauville, Arnaud Finite subgroups of PGL 2 (K), Vector bundles and complex geometry (Contemporary Mathematics), Volume 522, American Mathematical Society, 2010, pp. 23-29 | MR | Zbl

[2] Bray, John N.; Holt, Derek F.; Roney-Dougal, Colva M. The maximal subgroups of the low-dimensional finite classical groups, London Mathematical Society Lecture Note Series, 407, Cambridge University Press, 2013, xiv+438 pages | MR | Zbl

[3] Conway, John H.; Curtis, Robert T.; Norton, Simon P.; Parker, Richard A.; Wilson, Robert A Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups, Clarendon Press, 1985 | Zbl

[4] Coxeter, Harold S. M. The binary polyhedral groups and other generalizations of the quaternion group, Duke Math. J., Volume 7 (1940), pp. 367-379 | MR | Zbl

[5] Dickson, Leonard E. Linear Groups (with an exposition of the Galois field theory), B. G. Teubner, 1901 | Zbl

[6] Dornhoff, Larry Group Representation Theory. Part A: Ordinary Representation Theory, Pure and Applied Mathematics, 7, Marcel Dekker, 1971 | MR | Zbl

[7] Flicker, Yuval Z. Finite subgroups of SL(2,F ¯) and automorphy (preprint) | Zbl

[8] Flicker, Yuval Z. The finite subgroups of SL(3,F ¯) (preprint) | Zbl

[9] Flicker, Yuval Z. Linearly reductive finite subgroup schemes of SL(3) (preprint) | Zbl

[10] Gorenstein, Daniel; Lyons, Richard; Solomon, Ronald The Classification of the Finite Simple Groups. Part I, Chapter A: Almost simple K-groups, Mathematical Surveys and Monographs, 40, American Mathematical Society, 1998, xvi+419 pages | Zbl

[11] Huppert, Bertram Endliche Gruppen I, Grundlehren der mathematischen Wissenschaften, 134, Springer, 1967 | MR | Zbl

[12] Isaacs, I. Martin Character Theory of Finite Groups, Pure and Applied Mathematics, 69, Academic Press Inc., 1976, xii+304 pages | MR | Zbl

[13] Isaacs, I. Martin Finite Group Theory, Graduate Studies in Mathematics, 92, American Mathematical Society, 2008, xi+350 pages | MR | Zbl

[14] Mitchell, Howard H. Determination of the ordinary and modular ternary linear groups, American M. S. Trans., Volume 12 (2011), pp. 207-242 | DOI | MR | Zbl

[15] Serre, Jean-Pierre Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math., Volume 15 (1972), pp. 259-331 | DOI | Zbl

[16] Serre, Jean-Pierre Galois cohomology, Springer, 1997, x+210 pages | Zbl

[17] Serre, Jean-Pierre Finite Groups: An Introduction, Surveys of Modern Mathematics, 10, International Press., 2016, ix+179 pages | MR | Zbl

Cited by Sources: