The Sylow subgroups of the absolute Galois group Gal(Q)Gal(Q)

We describe the ℓ  -Sylow subgroups of Gal(Q)Gal(Q) for an odd prime ℓ  , by observing and studying their decomposition as F⋊ZℓF⋊Zℓ, where F is a free pro-ℓ   group, and ZℓZℓ are the ℓ  -adic integers. We determine the finite ZℓZℓ-quotients of F   and more generally show that every split embedding problem of ZℓZℓ-groups for F   is solvable. Moreover, we analyze the ZℓZℓ-action on generators of F.