TITLE:
Deep Transfers of p-Class Tower Groups
AUTHORS:
Daniel C. Mayer
KEYWORDS:
Hilbert p-Class Field Towers, p-Class Groups, p-Principalization, Quadratic Fields, Dihedral Fields of Degree 2p; Finite p-Groups, Two-Step Centralizers, Polarization Principle, Descendant Trees, p-Group Generation Algorithm, p-Multiplicator Rank, Relation Rank, Generator Rank, Deep Transfers, Shallow Transfers, Partial Order and Monotony Principle of Artin Patterns, Parametrized Polycyclic pc-Presentations, Commutator Calculus
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.6 No.1,
January
5,
2018
ABSTRACT: Let p be a prime. For any finite p-group G, the deep transfers T H,G ' : H / H ' → G ' / G " from the maximal subgroups H of index (G：H) = p in G to the derived subgroup G ' are introduced as an innovative tool for identifying G uniquely by means of the family of kernels ùd(G) =(ker(T H,G ')) (G： H) = p. For all finite 3-groups G of coclass cc(G) = 1, the family ùd(G) is determined explicitly. The results are applied to the Galois groups G =Gal(F3 (∞)/ F) of the Hilbert 3-class towers of all real quadratic fields F = Q(√d) with fundamental discriminants d > 1, 3-class group Cl3(F) □ C3 × C3, and total 3-principalization in each of their four unramified cyclic cubic extensions E/F. A systematic statistical evaluation is given for the complete range 1 d 7, and a few exceptional cases are pointed out for 1 d 8.