Mathematica Bohemica, Vol. 148, No. 2, pp. 149-180, 2023


Bicyclic commutator quotients with one non-elementary component

Daniel C. Mayer

Received August 25, 2021.   Published online May 3, 2022.

Abstract:  For any number field $K$ with non-elementary 3-class group ${\rm Cl}_3(K)\simeq C_{3^e}\times C_3$, $e\ge2$, the punctured capitulation type $\varkappa(K)$ of $K$ in its unramified cyclic cubic extensions $L_i$, $1\le i\le4$, 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 $G_2={\rm Gal}({\rm F}_3^2(K)/K)$ of the second Hilbert 3-class field of $K$. A classification of finite 3-groups $G$ with low order and bicyclic commutator quotient $G/G^\prime\simeq C_{3^e}\times C_3$, $2\le e\le6$, according to the algebraic invariant $\varkappa(G)$, admits conclusions concerning the length of the Hilbert 3-class field tower ${\rm F}_3^\infty(K)$ of imaginary quadratic number fields $K$.
Keywords:  Hilbert 3-class field tower; maximal unramified pro-3 extension; unramified cyclic cubic extensions; Galois action; imaginary quadratic fields; bicyclic 3-class group; punctured capitulation types; statistics; pro-3 groups; finite 3-groups; generator rank; relation rank; Schur $\sigma$-groups; low index normal subgroups; kernels of Artin transfers; abelian quotient invariants; $p$-group generation algorithm; descendant trees; antitony principle
Classification MSC:  11R37, 11R32, 11R11, 11R20, 11R29, 11Y40, 20D15, 20E18, 20E22, 20F05, 20F12, 20F14


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Affiliations:   Daniel C. Mayer, Naglergasse 53, 8010 Graz, Austria, e-mail: algebraic.number.theory@algebra.at


 
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