Finitely generated subgroups of free profinite groups and of some Galois groups (Q1282512)

A subgroup \(H\) of a profinite group \(G\) is called a ``composition subgroup'' if there exists a series of subgroups \[ G=N_0\supseteq N_1\supseteq\cdots\supseteq N_\lambda\supseteq N_{\lambda+1}\supseteq\cdots\supseteq N_\sigma=H, \] indexed by transfinite numbers, with the following properties: (i) \(N_{\lambda+1}\vartriangleleft N_\lambda\) for all \(\lambda<\sigma\); (ii) \(N_\mu=\bigcap_{\lambda<\mu}N_\lambda\) for limiting numbers \(\mu\leq\sigma\). When such a series is of finite length, the subgroup \(H\) is said to be ``subnormal''. With these definitions, the author proves essentially the following two theorems. Theorem 1. Let \(F\) be a free profinite or a free prosolvable group with \(\text{rank }F>1\). If a finitely generated subgroup \(H\) contains a nontrivial composition subgroup \(N\) of \(F\), then \(\text{rank }F<\infty\) and the index \((F:H)\) is finite. Theorem 2. Let \(K\) be a Hilbert field, and let one of the following conditions hold: 1) \(N\) is a nontrivial composition subgroup of the group \(G\) that is either the absolute Galois group \(\text{Gal}(K^{\text{sep}}/K)\) or the Galois group \(\text{Gal}(K^{\text{solv}}/K)\) of the maximal solvable extension over \(K\); 2) \(N\) is a nontrivial subnormal subgroup of the Galois group \(G=\text{Gal}(K^{(p)}/K)\) of the maximal \(p\)-extension over \(K\). In this case, \(N\) cannot be contained in any finitely generated subgroup of \(G\). Reviewer's remark: This last theorem is the correct translation of theorem 3 in the original Russian version.