We show that any infinite algebraic subgroup of the plane Cremona group over
a perfect field is contained in a maximal algebraic subgroup of the plane
Cremona group. We classify the maximal groups, and their subgroups of rational
points, up to conjugacy by a birational map.
For perfect fields k satisfying [k : k] > 2, we construct new normal subgroups of the plane Cremona group and provide an elementary proof of its non-simplicity, following the melody of the recent proof by Blanc, Lamy and Zimmermann that the Cremona group of rank n over (subfields of) the complex numbers is not simple for n 3.Résumé. -Pour les corps parfaits k qui satisfont [k : k] > 2, nous construisons de nouveaux sous-groupes distingués du groupe de Cremona du plan et nous donnons une preuve élémentaire de sa non-simplicité en suivant la mélodie de la preuve récente de Blanc, Lamy et Zimmermann du fait que le groupe de Cremona de rang n sur les (sous-corps des) nombres complexes n'est pas simple pour n 3.
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