Counting p-groups and nilpotent groups
Publications Mathématiques de l'IHÉS, Volume 92 (2000), p. 63-112
@article{PMIHES_2000__92__63_0,
     author = {Du Sautoy, Marcus},
     title = {Counting $p$-groups and nilpotent groups},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     publisher = {Institut des Hautes \'Etudes Scientifiques},
     volume = {92},
     year = {2000},
     pages = {63-112},
     zbl = {01656529},
     mrnumber = {2002f:11122},
     language = {en},
     url = {http://www.numdam.org/item/PMIHES_2000__92__63_0}
}
Du Sautoy, Marcus. Counting $p$-groups and nilpotent groups. Publications Mathématiques de l'IHÉS, Volume 92 (2000) pp. 63-112. http://www.numdam.org/item/PMIHES_2000__92__63_0/

[1] N. Blackburn, On a special class of p-groups, Acta Math. 100 (1958), 49-92. | MR 21 #1349 | Zbl 0083.24802

[2] R. M. Bryant and J. R. J. Groves, Algebraic groups of automorphisms of nilpotent groups and Lie algebras, J. London Math. Soc. 33 (1986), 453-466. | MR 87j:20082 | Zbl 0554.20008

[3] J. Denef, The rationality of the Poincaré series associated to the p-adic points on a variety, Invent. Math. 77 (1984), 1-23. | MR 86c:11043 | Zbl 0537.12011

[4] J. Denef and L. Van Den Dries, p-adic and real subanalytic sets, Annals of Math. 128 (1988), 79-138. | MR 89k:03034 | Zbl 0693.14012

[5] J. Denef and F. Loeser, Motivic Igusa zeta functions, J. Algebraic Geom., 7 (1998), 505-537. | MR 99j:14021 | Zbl 0943.14010

[6] J. Denef and F. Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math., 135 (1999), 201-232. | MR 99k:14002 | Zbl 0928.14004

[7] J. D. Dixon, M. P. F. Du Sautoy, A. Mann and D. Segal, Analytic pro-p groups, Second Edition, Cambridge Studies in Advanced Mathematics, 61, Cambridge, CUP, 1999. | MR 2000m:20039 | Zbl 0934.20001

[8] M. P. F. Du Sautoy, Finitely generated groups, p-adic analytic groups and Poincaré series, Annals of Math. 137 (1993), 639-670. | MR 94j:20029 | Zbl 0790.20044

[9] M. P. F. Du Sautoy, Zeta functions and counting finite p-groups, Electronic Research Announcements of the American Math. Soc., 5 (1999), 112-122. | MR 2000k:11102 | Zbl 1045.20503

[10] M. P. F. Du Sautoy, A nilpotent group and its elliptic curve : non-uniformity of local zeta functions of groups, MPI preprint 2000-2085. To appear in Israel J. of Math. 126. | Zbl 0998.20033

[11] M. P. F. Du Sautoy, Counting subgroups in nilpotent groups and points on elliptic curves, MPI preprint 2000-2086.

[12] M. P. F. Du Sautoy, Natural boundaries for zeta functions of groups, preprint.

[13] M. P. F. Du Sautoy and F. J. Grunewald, Analytic properties of Euler products of Igusa-type zeta functions and subgroup growth of nilpotent groups, C. R. Acad. Sci. Paris 329, Série 1 (1999), 351-356. | MR 2000g:11110 | Zbl 1062.11503

[14] M. P. F. Du Sautoy and F. J. Grunewald, Analytic properties of zeta functions and subgroup growth, Annals of Math. 152 (2000), 793-833. | MR 2002h:11084 | Zbl 1006.11051

[15] M. P. F. Du Sautoy and F. J. Grunewald, Uniformity for 2-generator free nilpotent groups, in preparation.

[16] M. P. F. Du Sautoy and F. Loeser, Motivic zeta functions of infinite dimensional Lie algebras, École polytechnique, preprint series 2000-2012.

[17] M. P. F. Du Sautoy and A. Lubotzky, Functional equations and uniformity for local zeta functions of nilpotent groups, Amer. J. Math. 118 (1996), 39-90. | MR 97g:11137 | Zbl 0864.20020

[18] M. P. F. Du Sautoy, J. J. Mcdermott and G. C. Smith, Zeta functions of crystallographic groups and analytic continuation, Proc. London Math. Soc. 79 (1999), 511-534. | MR 2000k:11103 | Zbl 1039.11056

[19] M. P. F. Du Sautoy and D. Segal, Zeta functions of groups, in New horizons in pro-p groups. Progress in Mathematics, vol. 184 (ed M. P. F. du Sautoy, D. Segal and A. Shalev), p. 249-286. Boston, Birkhäuser (2000). | MR 2001h:11123 | Zbl 01574587

[20] M. D. Fried and M. Jarden, Field Arithmetic, Springer-Verlag, Berlin, Heidelberg, New York, 1986. | MR 89b:12010 | Zbl 0625.12001

[21] F. J. Grunewald, D. Segal and G. C. Smith, Subgroups of finite index in nilpotent groups, Invent. Math. 93 (1988), 185-223. | MR 89m:11084 | Zbl 0651.20040

[22] G. Higman, Enumerating p-groups, I, Proc. London Math. Soc. 10 (1960), 24-30. | MR 22 #4779 | Zbl 0093.02603

[23] G. Higman, Enumerating p-groups, II, Proc. London Math. Soc. 10 (1960), 566-582. | MR 23 #A930 | Zbl 0201.36502

[24] K. Ireland and M. Rosen, A classical introduction to modern number theory, Second Edition, Graduate texts in mathematics 84, Springer-Verlag, New York, Berlin, Heidelberg, 1993.

[25] S. Lang, Algebra, Addison-Wesley, Reading, MA, 1965. | MR 33 #5416 | Zbl 0193.34701

[26] C. R. Leedham-Green and S. Mckay, On p-groups of maximal class II, Quart. J. Math. Oxford (2) 29 (1978), 175-186. | MR 58 #22281b | Zbl 0383.20016

[27] C. R. Leedham-Green and S. Mckay, On p-groups of maximal class III, Quart. J. Math. Oxford (2) 29 (1978), 281-299. | MR 58 #22281c | Zbl 0403.20013

[28] C. R. Leedham-Green and M. F. Newman, Space groups and groups of prime-power order I, Arch. Math. (Basel) 35 (1980), 193-202. | MR 81m:20029 | Zbl 0437.20016

[29] C. R. Leedham-Green, The structure of finite p-groups, J. London Math. Soc. 50 (1994), 49-67. | MR 95j:20022a | Zbl 0822.20018

[30] W. Magnus, A. Karras and D. Solitar, Combinatorial Group Theory, Wiley, Chichester, UK, 1966.

[31] M. F. Newman, Groups of prime-power order, Groups-Canberra 1989, Lecture Notes in Math., 1456 Springer-Verlag (1990), 49-62. | MR 91m:20031 | Zbl 0726.20011

[32] M. F. Newman and E. A. O'Brien, Classifying 2-groups by coclass, Trans. Amer. Math. Soc. 351 (1999), 131-169. | MR 99c:20020 | Zbl 0914.20020

[33] V. P. Platonov, The problem of strong approximation and the Kneser-Tits conjecture for algebraic groups, Math. USSR-Izv. 3 (1969), 1139-1147. | Zbl 0217.36301

[34] V. P. Platonov, Addendum, Math. USSR-Izv. 4 (1970), 784-786. | Zbl 0236.20034

[35] V. P. Platonov and A. S. Rapinchuk, Algebraic Groups and Number Theory, Pure and Applied Mathematics 139, London, Academic Press, 1994. | MR 95b:11039 | Zbl 0841.20046

[36] D. Segal, Polycyclic Groups, Cambridge tracts in mathematics, 82, CUP (1983). | MR 85h:20003 | Zbl 0516.20001

[37] A. Shalev, The structure of finite p-groups : effective proof of the coclass conjectures, Invent. Math. 115 (1994), 315-345. | MR 95j:20022b | Zbl 0795.20009

[38] C. C. Sims, Enumerating p-groups, Proc. London Math. Soc. 15 (1965), 151-166. | MR 30 #164 | Zbl 0133.28401

[39] R. P. Stanley, Enumerative Combinatorics, vol. 1, Cambridge Studies in Advanced Mathematics, 49, CUP, 1997. | Zbl 0889.05001