For a number field \(k\) and a prime number \(p\), the \(p\)-class field tower of \(k\) is the sequence of fields \(k=L^{(0)}(k)\subseteq L^{(1)}(k)\subseteq\cdots \subseteq L^{(i)}(k)\subseteq\cdots \subseteq \bigcup_{i\geq 0}L^{(i)}(k)=:\mathcal{L}(k)\), where \(L(k)=L^{(1)}(k)\) is the Hilbert \(p\)-class field of \(k\) and, for \(i\in\mathbb{N}\), \(L^{(i+1)}(k)\) is the Hilbert \(p\)-class of \(L^{(i)}(k)\). The field \(\mathcal{L}(k)\) is called the maximal unramified pro-\(p\)-extension of \(k\). We know, by a Theorem of E. S. Golod and I. R. Shafarevich [Izv. Akad. Nauk SSSR, Ser. Mat. 28, 261–272 (1964; Zbl 0136.02602)], that Gal\((\mathcal{L}(k)/k)\) can be infinite. For a \(\mathbb{Z}_p\)-extension \(k_{\infty}\) of \(k\), let \(A(k_n)\) be the \(p\)-Sylow subgroup of the ideal class group of the \(n\)-th layer \(k_n\). The Iwasawa module for \(k_{\infty}/k\) is the inverse limit \(X=\lim\limits_{\leftarrow n} A(k_n)\) with respect to the norm maps.
The authors consider the Galois group of the maximal unramified pro-\(2\)-extension of \(k_{\infty}\): \(G:=\)Gal\((\mathcal{L}(k_{\infty})/k_{\infty})\) that is isomorphic to \(\lim\limits_{\leftarrow n}\)Gal\(( \mathcal{L}(k_n)/k_n)\) with respect to the restriction maps. We have that \(G/G'\cong \)Gal\((L(k_{\infty})/k_{\infty})\cong X\), where \(G'\) is the commutator subgroup of \(G\).
The paper first studies necessary and sufficient conditions for a finite \(2\)-group \(G\) whose abelianization is of type \((2^n,2^m)\), with \(n\geq 2\) and \(m\geq 2\) to be abelian (resp. metacyclic non-abelian) or not. The tools used to this end are, the transfer map, results about the problem of capitulation and the ranks of maximal subgroups of \(G\). The main result for this part is the following. Let \(G\) be a finite \(2\)-group such that \(G/G'\) is of type \((2^n,2^m)\) with \(n,m\geq 2\). Let \(H_i\), \(i\in\{1,2,3\}\) be the three maximal subgroups of \(G\). Then: (1) \(G\) is abelian if and only if \(d(H_i)=2\) and \(\#\ker \)Ver\(_{G\to H_i}=4\) for all \(i\); (2) \(G\) is metacyclic non-abelian if and only if \(d(H_i)=2\) and \(\# \ker \)Ver\(_{G\to H_i}\leq 2\) for all \(i\); (3) \(G\) is non-metacyclic non-abelian if and only if \(d( H_i)=3\) for some \(i\in\{1,2,3\}\). In this case \(d(H_i)=3\) for all \(i\). Here Ver\(_{G\to H}\colon G/G'\to H/H'\) is the transfer map and the rank \(d(H)\) of a \(2\)-group is the dimension of \(G/\Phi(G)\) over \(\mathbb{Z}/2\mathbb{Z}=\mathbb{F}_2\), where \(\Phi(H)\) is the Frattini subgroup of \(H\).
Next, in the second part of the paper, the authors study the structure of \(G=\)Gal\((\mathcal{L}(k_{\infty})/k_{\infty})\) for the family of real quadratic number fields \(k=\mathbb{Q} \big(\sqrt{q_1q_2q_3}\big)\), where \(q_i\equiv 3\bmod 4\) are distinct prime integers. The result obtained is that \(G\) is an abelian pro-\(2\)-group of rank \(2\).