On prosolvable subgroups of profinite free products and some applications (Q1344934)

Let \(G\) be a profinite group, and \(\mathfrak G\) a family of closed subgroups of \(G\) closed under the conjugate action of \(G\). A finite \({\mathfrak G}\)- embedding problem for \(G\) is a diagram \(EP_{\mathfrak G}=(\gamma:G\to A,\alpha:B\to A,{\mathfrak B},\{\beta_\Gamma\}_{\Gamma\in{\mathfrak G}})\), where \(A\), \(B\) are finite groups, \(\gamma\), \(\alpha\) are surjective homomorphisms, and \(\mathfrak B\) is a conjugacy-closed family of subgroups of \(B\) such that every \(\Gamma\in{\mathfrak G}\) has a homomorphism into a member of \(\mathfrak B\) lifting \(\gamma\). If there exists \(\beta:G\to B\) with \(\gamma=\alpha\beta\), then \(\beta\) is called a solution of \(EP_{\mathfrak G}\), and if moreover \(\beta(\mathfrak G)\subset{\mathfrak B}\) holds, \(\beta\) is called a strong solution. We say that \(G\) is [strongly] \(\mathfrak G\)- projective if every finite \(\mathfrak G\)-embedding problem for \(G\) has a [strong] solution. It is shown that, under some technical conditions, any not-too-small prosolvable (or any finite) subgroup of a strongly \(\mathfrak G\)-projective group \(G\) must be contained in some member of \(\mathfrak G\) (Theorem 2). The author also applies this result to show that a certain general hypothesis on \(\mathfrak G\) is sufficient for a \(\mathfrak G\)-projective group to be strongly \(\mathfrak G\)-projective (Theorem 3). This paper provides a step to another article of the author: ``Embedding problems over large fields'' [Ann. Math., II. Ser. (to appear)].