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 trees of finite p-groups.
References
[1]
du Sautoy, M. (2001) Counting p-Groups and Nilpotent Groups. Publications Mathématiques de l’Institut des Hautes études Scientifiques, 92, 63-112.
http://dx.doi.org/10.1007/BF02698914
[2]
Eick, B. and Leedham-Green, C. (2008) On the Classification of Prime-Power Groups by Coclass. Bulletin of the London Mathematical Society, 40, 274-288.
http://dx.doi.org/10.1112/blms/bdn007
[3]
Artin, E. (1929) Idealklassen in Oberkorpern und allgemeines Reziprozitatsgesetz. Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg, 7, 46-51.
http://dx.doi.org/10.1007/BF02941159
[4]
Mayer, D.C. (2012) Transfers of Metabelian p-Groups. Monatshefte für Mathematik, 166, 467-495.
http://dx.doi.org/10.1007/s00605-010-0277-x
[5]
Mayer, D.C. (2014) Principalization Algorithm via Class Group Structure. Journal de Théorie des Nombres de Bordeaux, 26, 415-464.
[6]
Mayer, D.C. (2013) The Distribution of Second p-Class Groups on Coclass Graphs. Journal de Théorie des Nombres de Bordeaux, 25, 401-456.
[7]
Bush, M.R. and Mayer, D.C. (2015) 3-Class Field Towers of Exact Length 3. Journal of Number Theory, 147, 766-777.
http://dx.doi.org/10.1016/j.jnt.2014.08.010
[8]
Newman, M.F. (1977) Determination of Groups of Prime-Power Order. In: Bryce, R.A., Cossey, J., Newman, M.F., Eds., Group Theory, Springer, Berlin, 73-84.
[9]
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
[10]
Holt, D.F., Eick, B. and O’Brien, E.A. (2005) Handbook of Computational Group Theory, Discrete Mathematics and Its Applications. Chapman and Hall/CRC Press, Boca Raton.
http://dx.doi.org/10.1201/9781420035216
[11]
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.
[12]
The GAP Group (2014) GAP—Groups, Algorithms, and Programming—A System for Computational Discrete Algebra. Version 4.7.5, Aachen, Braunschweig, Fort Collins, St. Andrews.
http://www.gap-system.org
[13]
The MAGMA Group (2014) MAGMA Computational Algebra System. Version 2.21-1, Sydney.
http://magma.maths.usyd.edu.au
[14]
Ascione, J.A., Havas, G. and Leedham-Green, C.R. (1977) A Computer Aided Classification of Certain Groups of Prime Power Order. Bulletin of the Australian Mathematical Society, 17, 257-274.
http://dx.doi.org/10.1017/s0004972700010467
[15]
Nebelung, B. (1989) Klassifikation metabelscher 3-Gruppen mit Faktorkommutatorgruppe vom Typ (3,3) und Anwendung auf das Kapitulationsproblem. Inaugural Dissertation, Universitat zu Koln, Koln.
[16]
Besche, H.U., Eick, B. and O’Brien, E.A. (2002) A Millennium Project: Constructing Small Groups. International Journal of Algebra and Computation, 12, 623-644.
http://dx.doi.org/10.1142/S0218196702001115
[17]
Besche, H.U., Eick, B. and O’Brien, E.A. (2005) The Small Groups Library—A Library of Groups of Small Order. An Accepted and Refereed GAP 4 Package, Available also in MAGMA.
[18]
Ascione, J.A. (1979) On 3-Groups of Second Maximal Class. Ph.D. Thesis, Australian National University, Canberra.
[19]
Ascione, J.A. (1980) On 3-Groups of Second Maximal Class. Bulletin of the Australian Mathematical Society, 21, 473-474.
http://dx.doi.org/10.1017/S0004972700006298
[20]
Newman, M.F. (1990) Groups of Prime-Power Order. Lecture Notes in Mathematics, 1456, 49-62.
http://dx.doi.org/10.1007/BFb0100730
[21]
Leedham-Green, C.R. and Newman, M.F. (1980) Space Groups and Groups of Prime Power Order I. Archiv der Mathematik, 35, 193-203.
http://dx.doi.org/10.1007/BF01235338
[22]
du Sautoy, M. and Segal, D. (2000) Zeta Functions of Groups. In: du Sautoy, M., Segal, D. and Shalev, A., Eds., New Horizons in Pro-p Groups, Progress in Mathematics, Birkhauser, Basel, 249-286.
[23]
Leedham-Green, C.R. and McKay, S. (2002) The Structure of Groups of Prime Power Order, London Mathematical Society Monographs. New Series 27, Oxford University Press, Oxford.
[24]
Eick, B., Leedham-Green, C.R., Newman, M.F. and O’Brien, E.A. (2013) On the Classification of Groups of Prime-Power Order by Coclass: The 3-Groups of Coclass 2. International Journal of Algebra and Computation, 23, 1243-1288.
http://dx.doi.org/10.1142/s0218196713500252
[25]
Newman, M.F. and O’Brien, E.A. (1999) Classifying 2-Groups by Coclass. Transactions of the American Mathematical Society, 351, 131-169.
http://dx.doi.org/10.1090/S0002-9947-99-02124-8
[26]
Dietrich, H., Eick, B. and Feichtenschlager, D. (2008) Investigating p-Groups by Coclass with GAP. In: Kappe, L.-C., Magidin, A. and Morse, R.F., Eds., Computational Group Theory and the Theory of Groups, AMS, Providence, 45-61.
http://dx.doi.org/10.1090/conm/470/09185
[27]
Shalev, A. (1994) The Structure of Finite p-Groups: Effective Proof of the Coclass Conjectures. Inventiones Mathematicae, 115, 315-345.
http://dx.doi.org/10.1007/BF01231763
[28]
Leedham-Green, C.R. (1994) The Structure of Finite p-Groups. Journal London Mathematical Society, 50, 49-67.
http://dx.doi.org/10.1112/jlms/50.1.49
[29]
Hall, M. and Senior, J.K. (1964) The Groups of Order 2n (n ≤ 6). Macmillan, New York.
[30]
Hall, P. (1940) The Classification of Prime-Power Groups. Journal für Die Reine und Angewandte Mathematik, 182, 130-141.
[31]
Blackburn, N. (1958) On a Special Class of p-Groups. Acta Mathematica, 100, 45-92.
http://dx.doi.org/10.1007/BF02559602
[32]
Taussky, O. (1937) A Remark on the Class Field Tower. Journal London Mathematical Society, 12, 82-85.
http://dx.doi.org/10.1112/jlms/s1-12.1.82
[33]
Bagnera, G. (1898) La composizione dei gruppi finiti il cui grado è la quinta potenza di un numero primo. Annali di Matematica Pura ed Applicata, 1, 137-228.
http://dx.doi.org/10.1007/bf02419191
[34]
Arrigoni, M. (1998) On Schur σ-Groups. Mathematische Nachrichten, 192, 71-89.
http://dx.doi.org/10.1002/mana.19981920105
[35]
Boston, N., Bush, M.R. and Hajir, F. (2015) Heuristics for p-Class Towers of Imaginary Quadratic Fields. Mathematische Annalen, in Press.
[36]
Koch, H. and Venkov, B.B. (1975) über den p-Klassenkorperturm eines imaginar-quadratischen Zahlkorpers. Astérisque, 24-25, 57-67.
[37]
Benjamin, E., Lemmermeyer, F. and Snyder, C. (2003) Imaginary Quadratic Fields with Cl2(k)(2,2,2). Journal of Number Theory, 103, 38-70.
http://dx.doi.org/10.1016/S0022-314X(03)00084-2
[38]
Shafarevich, I.R. (1964) Extensions with Prescribed Ramification Points (Russian). Publications Mathématiques de l’IHéS, 18, 71-95.
[39]
Boston, N. and Nover, H. (2006) Computing Pro-p Galois Groups. Algorithmic Number Theory: Lecture Notes in Computer Science, 4076, 1-10.
[40]
Mayer, D.C. (2012) The Second p-Class Group of a Number Field. International Journal of Number Theory, 8, 471-505. http://dx.doi.org/10.1142/S179304211250025X
[41]
Nover, H. (2009) Computation of Galois Groups of 2-Class Towers. Ph.D. Thesis, University of Wisconsin, Madison.
[42]
Azizi, A., Zekhnini, A. and Taous, M. (2015) Coclass of for Some Fields with 2-Class Groups of Type (2,2,2). Journal of Algebra and Its Applications, in Press.
[43]
Bosma, W., Cannon, J. and Playoust, C. (1997) The Magma Algebra System. I. The User Language. Journal of Symbolic Computation, 24, 235-265.
http://dx.doi.org/10.1006/jsco.1996.0125
[44]
Bosma, W., Cannon, J.J., Fieker, C. and Steels, A., Eds. (2014) Handbook of Magma Functions. Edition 2.21, University of Sydney, Sydney.
[45]
Mayer, D.C. and Newman, M.F. (2013) Finite 3-Groups as Viewed from Class Field Theory, Groups St. Andrews 2013. University of St. Andrews, Fife, Scotland.
[46]
Mayer, D.C., Bush, M.R. and Newman, M.F. (2013) 3-Class Field Towers of Exact Length 3. 18th OMG Congress and 123rd Annual DMV Meeting 2013, University of Innsbruck, Tyrol, Austria.
[47]
Mayer, D.C., Bush, M.R. and Newman, M.F. (2013) Class Towers and Capitulation over Quadratic Fields. West Coast Number Theory 2013, Asilomar Conference Center, Pacific Grove.
[48]
Scholz, A. and Taussky, O. (1934) Die Hauptideale der kubischen Klassenkorper imaginar quadratischer Zahlkorper: Ihre rechnerische Bestimmung und ihr Einflu auf den Klassenkorperturm. Journal Für Die Reine und Angewandte Mathematik, 171, 19-41.
[49]
Heider, F.-P. and Schmithals, B. (1982) Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen. Journal Für Die Reine und Angewandte Mathematik, 336, 1-25.