Periodic Sequences of p-Class Tower Groups

Daniel C. Mayer

Austrian Science Fund.

DOI: 10.4236/jamp.2015.37090                

Abstract

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.

Keywords

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

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Mayer, D.C. (2015) Periodic Bifurcations in Descendant Trees of Finite p-Groups. Advances in Pure Mathematics, 5, 162-195. http://dx.doi.org/10.4236/apm.2015.54020 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. http://dx.doi.org/10.1007/bfb0087814
[4]	O’Brien, E.A. (1990) The p-Group Generation Algorithm. Journal of Symbolic Computation, 9, 677-698. http://dx.doi.org/10.1016/S0747-7171(08)80082-X
[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. http://dx.doi.org/10.1142/S0218196702001115
[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. http://www.gap-system.org
[9]	Bosma, W., Cannon, J. and Playoust, C. (1997) The Magma Algebra System. I. The User Language. J. Symbolic Comput. 24, 235-265. http://dx.doi.org/10.1006/jsco.1996.0125
[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. http://magma.maths.usyd.edu.au
[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. http://dx.doi.org/10.1007/BF01446682
[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. http://dx.doi.org/10.1007/BF02941159
[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. http://oeis.org/