TY - JOUR

T1 - Conjugacy classes of finite subgroups of SL(2, F ), SL(3, FS)

AU - Flicker, Yuval Z.

N1 - Publisher Copyright:
© Société Arithmétique de Bordeaux, 2019, tous droits réservés.

PY - 2019

Y1 - 2019

N2 - 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 = Fs 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, Fs). 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, Fs), that is, the minimal field F1 with FĎ1 = Fs such that the finite group G embeds in SL(2, F1).

AB - 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 = Fs 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, Fs). 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, Fs), that is, the minimal field F1 with FĎ1 = Fs such that the finite group G embeds in SL(2, F1).

KW - Algebraic classification

KW - Galois cohomology

KW - Rational classification

KW - Representation theory

KW - SL(2, F)

KW - SL(3, F)

UR - http://www.scopus.com/inward/record.url?scp=85086774662&partnerID=8YFLogxK

U2 - 10.5802/jtnb.1094

DO - 10.5802/jtnb.1094

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AN - SCOPUS:85086774662

SN - 1246-7405

VL - 31

SP - 555

EP - 571

JO - Journal de Theorie des Nombres de Bordeaux

JF - Journal de Theorie des Nombres de Bordeaux

IS - 3

ER -