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Deep Transfers of p-Class Tower Groups

Journal of Applied Mathematics and Physics 2018年第1期6卷 36-50页

作者： Daniel C. Mayer Naglergasse 53 Graz Austria

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.

关键词： 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

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Co-Periodicity Isomorphisms between Forests of Finite <I>p</I>-Groups

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Advances in Pure Mathematics 2018年第1期8卷 77-140页

作者： Daniel C. Mayer Naglergasse 53 Graz Austria

Based on a general theory of descendant trees of finite p-groups and the virtual periodicity isomorphisms between the branches of a coclass subtree, the behavior of algebraic invariants of the tree vertices and their automorphism groups under these isomorphisms is described with simple transformation laws. For the tree of finite 3-groups with elementary bicyclic commutator qu-otient, the information content of each coclass subtree with metabelian main-line is shown to be finite. As a striking novelty in this paper, evidence is provided of co-periodicity isomorphisms between coclass forests which reduce the information content of the entire metabelian skeleton and a significant part of non-metabelian vertices to a finite amount of data.

关键词： Finite p-Groups Descendant Trees Pro-p Groups Coclass Forests Generator Rank Relation Rank Nuclear Rank parametrized Polycyclic Pc-Presentations Automorphism Groups Central Series Two-Step Centralizers Commutator Calculus Transfer Kernels Abelian Quotient Invariants p-Group Generation Algorithm

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Periodic Sequences of p-Class Tower Groups

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Journal of Applied Mathematics and Physics 2015年第7期3卷 746-756页

作者： Daniel C. Mayer Austrian Science Fund

Recent examples of periodic bifurcations in descendant trees of finite p-groups with ?are used to show that the possible p-class tower groups G of certain multiquadratic fields K with p- class group of type (2,2,2) , resp. (3,3), form periodic sequences in the descendant tree of the elementary Abelian root , resp. . The particular vertex of the periodic sequence which occurs as the p-class tower group G of an assigned field K is determined uniquely by the p-class number of a quadratic, resp. cubic, auxiliary field k, associated unambiguously to K. Consequently, the hard problem of identifying the p-class tower group G is reduced to an easy computation of low degree arithmetical invariants.

关键词： p-Class Field Towers p-Principalization p-Class Groups Quadratic Fields Multiquadratic Fields Cubic Fields Finite p-Groups parametrized Pc-Presentations p-Group Generation Algorithm

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Periodic Bifurcations in Descendant Trees of Finite <i>p</i>-Groups

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Advances in Pure Mathematics 2015年第4期5卷 162-195页

作者： Daniel C. Mayer Naglergasse 53 8010 Graz Austria

Theoretical background and an implementation of the p-group generation algorithm by Newman and O’Brien are used to provide computational evidence of a new type of periodically repeating patterns in pruned descendant ... 详细信息

关键词： Finite p-Group Central Series Descendant Tree Pro-p Group Coclass Tree p-Covering Group Nuclear Rank Multifurcation Coclass Graph parametrized Presentation Commutator Calculus Schur -Group

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