Bicyclic commutator quotients with one non-elementary component

Daniel Mayer

Mathematica Bohemica (2023)

  • Volume: 148, Issue: 2, page 149-180
  • ISSN: 0862-7959

Abstract

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For any number field K with non-elementary 3 -class group Cl 3 ( K ) ≃ C 3 e × C 3 , e ≥ 2 , the punctured capitulation type ϰ ( K ) of K in its unramified cyclic cubic extensions L i , 1 ≤ i ≤ 4 , is an orbit under the action of S 3 × S 3 . By means of Artin’s reciprocity law, the arithmetical invariant ϰ ( K ) is translated to the punctured transfer kernel type ϰ ( G 2 ) of the automorphism group G 2 = Gal ( 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 ' ≃ C 3 e × C 3 , 2 ≤ e ≤ 6 , according to the algebraic invariant ϰ ( G ) , admits conclusions concerning the length of the Hilbert 3 -class field tower F 3 ∞ ( K ) of imaginary quadratic number fields K .

How to cite

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Mayer, Daniel. "Bicyclic commutator quotients with one non-elementary component." Mathematica Bohemica 148.2 (2023): 149-180. <http://eudml.org/doc/299567>.

@article{Mayer2023,
abstract = {For any number field $K$ with non-elementary $3$-class group $\{\rm Cl\}_3(K)\simeq C_\{3^e\}\times C_3$, $e\ge 2$, 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 $G_2=\{\rm Gal\}(\{\rm F\}_3^2(K)/K)$ of the second Hilbert $3$-class field of $K$. A classification of finite $3$-groups $G$ with low order and bicyclic commutator quotient $G/G^\prime \simeq C_\{3^e\}\times C_3$, $2\le e\le 6$, 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 $K$.},
author = {Mayer, Daniel},
journal = {Mathematica Bohemica},
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},
language = {eng},
number = {2},
pages = {149-180},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Bicyclic commutator quotients with one non-elementary component},
url = {http://eudml.org/doc/299567},
volume = {148},
year = {2023},
}

TY - JOUR
AU - Mayer, Daniel
TI - Bicyclic commutator quotients with one non-elementary component
JO - Mathematica Bohemica
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 148
IS - 2
SP - 149
EP - 180
AB - For any number field $K$ with non-elementary $3$-class group ${\rm Cl}_3(K)\simeq C_{3^e}\times C_3$, $e\ge 2$, 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 $G_2={\rm Gal}({\rm F}_3^2(K)/K)$ of the second Hilbert $3$-class field of $K$. A classification of finite $3$-groups $G$ with low order and bicyclic commutator quotient $G/G^\prime \simeq C_{3^e}\times C_3$, $2\le e\le 6$, 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 $K$.
LA - eng
KW - 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
UR - http://eudml.org/doc/299567
ER -

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