Deep Transfers of p-Class Tower Groups - Journal of Applied Mathematics and Physics

JAMP > Vol.6 No.1, January 2018

Deep Transfers of p-Class Tower Groups ()

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DOI: 10.4236/jamp.2018.61005 635 Downloads 1,221 Views Citations

Author(s)

Daniel C. Mayer

Affiliation(s)

Naglergasse 53, Graz, Austria.

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 ù_d(G) =(ker(T _{H,G^')) _{(G： H) = p}. For all finite 3-groups G of coclass cc(G) = 1, the family ù_d(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 < 10⁷, and a few exceptional cases are pointed out for 1 < d < 10⁸.}}