Bicyclic commutator quotients with one non-elementary component

• Published in: Mathematica bohemica Vol. 148; no. 2; pp. 149 - 180
• Main Author: Daniel C. Mayer
• Format: Journal Article
• Language: English
• Published: Institute of Mathematics of the Czech Academy of Science 01.07.2023
• Subjects: abelian quotient invariants; antitony principle; bicyclic $3$-class group; descendant trees; finite $3$-groups; galois action; generator rank; hilbert $3$-class field tower; imaginary quadratic fields; kernels of artin transfers; low index normal subgroups; maximal unramified pro-$3$ extension; p$-group generation algorithm; pro-$3$ groups; punctured capitulation types; relation rank; schur $\sigma$-groups; statistics; unramified cyclic cubic extensions
• ISSN: 0862-7959; 2464-7136
• DOI: 10.21136/MB.2022.0127-21

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 $K$.