Periodic Sequences of p-Class Tower Groups
Daniel C. Mayer
Austrian Science Fund.
DOI: 10.4236/jamp.2015.37090   PDF   HTML   XML   3,011 Downloads   3,534 Views   Citations


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.

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Mayer, D. (2015) Periodic Sequences of p-Class Tower Groups. Journal of Applied Mathematics and Physics, 3, 746-756. doi: 10.4236/jamp.2015.37090.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Mayer, D.C. (2015) Periodic Bifurcations in Descendant Trees of Finite p-Groups. Advances in Pure Mathematics, 5, 162-195. Special Issue on Group Theory, March 2015. (arXiv: 1502.03390v1 [math.GT] 11 Feb 2015.)
[2] Mayer, D.C. (2015) Index-p Abelianization data of p-Class Tower Groups, to appear in Advances in Pure Mathematics, 5, Special Issue on Number Theory and Cryptography, April 2015. (arXiv: 1502.03388v1 [math.NT] 11 Feb 2015.)
[3] Newman, M.F. (1977) Determination of Groups of Prime-Power Order. In: Lecture Notes in Math., vol. 573, Group Theory, Canberra, Springer, Berlin, 73-84.
[4] O’Brien, E.A. (1990) The p-Group Generation Algorithm. Journal of Symbolic Computation, 9, 677-698.
[5] Besche, H.U., Eick, B. and O’Brien, E.A. (2002) A Millen-nium Project: Constructing Small Groups. Int. J. Algebra Comput., 12, 623-644.
[6] Besche, H.U., Eick, B. and O’Brien, E.A. (2005) The SmallGroups Library—A Library of Groups of Small Order. An accepted and refereed GAP 4 package, available also in MAGMA.
[7] Gamble, G., Nickel, W. and O’Brien, E.A. (2006) ANU p-Quotient—p-Quotient and p-Group Generation Algorithms. An accepted GAP 4 package, available also in MAGMA.
[8] The GAP Group (2015) GAP—Groups, Algorithms, and Programming—a System for Computational Discrete Algebra. Version 4.7.7, Aachen, Braunschweig, Fort Collins, St. Andrews.
[9] Bosma, W., Cannon, J. and Playoust, C. (1997) The Magma Algebra System. I. The User Language. J. Symbolic Comput. 24, 235-265.
[10] Bosma, W., Cannon, J.J., Fieker, C. and Steels, A. (eds.) (2015) Hand-book of Magma Functions. Edition 2.21, Univ. of Sydney, Sydney.
[11] The MAGMA Group (2015) MAGMA Computational Algebra System. Version 2.21-2, Sydney.
[12] Bush, M.R. and Mayer, D.C. (2015) 3-Class Field Towers of Exact Length 3. J. Number Theory, 147, 766-777. (arXiv: 1312.0251v1 [math.NT] 1 Dec 2013.)
[13] Hilbert, D. (1894) Ueber den Di-richlet'schen biquadratischen Zahlk?rper. Mathematische Annalen, 45, 309-340.
[14] Azizi, A., Zekhnini, A. and Taous, M. (2015) Coclass of for Some Fields with 2-Class Groups of Type . To appear in J. Algebra Appl.
[15] Artin, E. (1929) Idealklassen in Oberk?rpern und allgemeines Reziprozit?tsgesetz. Abh. Math. Sem. Univ. Hamburg, 7, 46-51.
[16] Mayer, D.C. (2013) The Distribution of Second p-Class Groups on Coclass Graphs. J. Théor. Nombres Bordeaux, 25, 401-456. (27th Journées Arithmétiques, Faculty of Mathematics and In-formatics, Univ. of Vilnius, Lithuania, 2011.)
[17] Mayer, D.C. (2012) The Second p-Class Group of a Number Field. Int. J. Number Theory, 8, 471-505.
[18] Mayer, D.C. (2014) Principalization Algorithm via Class Group Structure. J. Théor. Nombres Bordeaux, 26, 415-464.
[19] Boston, N., Bush, M.R. and Hajir, F. (2015) Heuristics for p-Class Towers of Imaginary Quadratic Fields. To appear in Math. Annalen. (arXiv: 1111.4679v2 [math.NT] 10 Dec 2014.)
[20] Fieker, C. (2001) Computing Class Fields via the Artin Map. Math. Comp., 70, 1293-1303.
[21] Sloane, N.J.A. (2014) The On-Line Encyclopedia of Integer Sequences (OEIS). The OEIS Foundation Inc.

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