Geodesic
Finitely generated subgroups of free profinite groups and of some Galois groups
Matematičeskie zametki, Tome 64 (1998) no. 1, pp. 95-106

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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 $[F:H]\infty$. 
@article{MZM_1998_64_1_a9,
     author = {O. V. Mel'nikov},
     title = {Finitely generated subgroups of free profinite groups and of some {Galois} groups},
     journal = {Matemati\v{c}eskie zametki},
     pages = {95--106},
     publisher = {mathdoc},
     volume = {64},
     number = {1},
     year = {1998},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1998_64_1_a9/}
}
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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/