Geodesic
Finitely generated subgroups of free profinite groups and of some Galois groups
Matematičeskie zametki, Tome 64 (1998) no. 1, pp. 95-106 
O. V. Mel'nikov. Finitely generated subgroups of free profinite groups and of some Galois groups. Matematičeskie zametki, Tome 64 (1998) no. 1, pp. 95-106. http://geodesic.mathdoc.fr/item/MZM_1998_64_1_a9/
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     author = {O. V. Mel'nikov},
     title = {Finitely generated subgroups of free profinite groups and of some {Galois} groups},
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     pages = {95--106},
     year = {1998},
     volume = {64},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1998_64_1_a9/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

It is proved that, for a (closed) subgroup $H$ of a free profinite or free prosolvable group $F$ of $\operatorname{rank}F>1$ such that $H$ contains a nontrivial composition subgroup $N$ of $F$, we have $\operatorname{rank}F<\infty$ and $[F:H]<\infty$.

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