Bicyclic commutator quotients with one non-elementary component

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