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We give an infinitesimal criterion, in the analytic setting, for a vectorspace to be locally homogeneous under some group action. Our approach differsfrom those which resort to an inverse function theorem (e.g. those of Moser,Zehnder or Sergeraert), because we use the underlying group structure in anessential way. In particular, this allows to replace the estimate of theinverse map of the Lie algebra action at an arbitrary tangent plane, by anestimate of the vectors tangent at the origin. Our proof relies on theiterative method of Kolmogorov and Arnold in their proof of the invariant toritheorem. The theorem of this note will be used in subsequent works.

Un th\'eor\`eme sur les actions de groupes de dimension infinie