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Bicyclic commutator quotients with one non-elementary component. (English) Zbl 07729570

Summary: For any number field \(K\) with non-elementary \(3\)-class group \(\mathrm{Cl}_3(K)\simeq C_{3^e}\times C_3\), \(e\ge 2\), the punctured capitulation type \(\varkappa(K)\) of \(K\) in its unramified cyclic cubic extensions \(L_i\), \(1\le i\le 4\), is an orbit under the action of \(S_3\times S_3\). By means of Artin’s reciprocity law, the arithmetical invariant \(\varkappa(K)\) is translated to the punctured transfer kernel type \(\varkappa(G_2)\) of the automorphism group \(G_2=\mathrm{Gal}(\mathrm{F}_3^2(K)/K)\) of the second Hilbert \(3\)-class field of \(K\). A classification of finite \(3\)-groups \(G\) with low order and bicyclic commutator quotient \(G/G^\prime\simeq C_{3^e}\times C_3\), \(2\le e\le 6\), according to the algebraic invariant \(\varkappa(G)\), admits conclusions concerning the length of the Hilbert \(3\)-class field tower \(\mathrm{F}_3^\infty(K)\) of imaginary quadratic number fields \(K\).

MSC:

11R37 Class field theory
11R32 Galois theory
11R11 Quadratic extensions
11R20 Other abelian and metabelian extensions
11R29 Class numbers, class groups, discriminants
11Y40 Algebraic number theory computations
20D15 Finite nilpotent groups, \(p\)-groups
20E18 Limits, profinite groups
20E22 Extensions, wreath products, and other compositions of groups
20F05 Generators, relations, and presentations of groups
20F12 Commutator calculus
20F14 Derived series, central series, and generalizations for groups
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