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

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).