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Enumerating classes and characters of $ p$-groups


Authors: E. A. O’Brien and C. Voll
Journal: Trans. Amer. Math. Soc. 367 (2015), 7775-7796
MSC (2010): Primary 20C15, 20D15
DOI: https://doi.org/10.1090/tran/6276
Published electronically: April 3, 2015
MathSciNet review: 3391899
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Abstract: We develop general formulae for the numbers of conjugacy classes and irreducible complex characters of finite $ p$-groups of nilpotency class less than $ p$. This allows us to unify and generalize a number of existing enumerative results, and to obtain new such results for generalizations of relatively free $ p$-groups of exponent $ p$. Our main tools are the Lazard correspondence and the Kirillov orbit method.


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  • [1] Nir Avni, Benjamin Klopsch, Uri Onn, and Christopher Voll, Representation zeta functions of compact $ p$-adic analytic groups and arithmetic groups, Duke Math. J. 162 (2013), no. 1, 111-197. MR 3011874, https://doi.org/10.1215/00127094-1959198
  • [2] Edward A. Bender, On Buckheister's enumeration of $ n$ $ \times \ n$ matrices, J. Combinatorial Theory Ser. A 17 (1974), 273-274. MR 0347622 (50 #125)
  • [3] Nigel Boston and I. M. Isaacs, Class numbers of $ p$-groups of a given order, J. Algebra 279 (2004), no. 2, 810-819. MR 2078943 (2005f:20034), https://doi.org/10.1016/j.jalgebra.2004.03.006
  • [4] Mitya Boyarchenko, Representations of unipotent groups over local fields and Gutkin's conjecture, Math. Res. Lett. 18 (2011), no. 3, 539-557. MR 2802587 (2012d:22021), https://doi.org/10.4310/MRL.2011.v18.n3.a14
  • [5] Mitya Boyarchenko and Maria Sabitova, The orbit method for profinite groups and a $ p$-adic analogue of Brown's theorem, Israel J. Math. 165 (2008), 67-91. MR 2403615 (2009b:20046), https://doi.org/10.1007/s11856-008-1004-3
  • [6] L. Carlitz and John H. Hodges, Distribution of bordered symmetric, skew and hermitian matrices in a finite field, J. Reine Angew. Math. 195 (1955), 192-201 (1956). MR 0075983 (17,828f)
  • [7] John Cossey and Trevor Hawkes, Sets of $ p$-powers as conjugacy class sizes, Proc. Amer. Math. Soc. 128 (2000), no. 1, 49-51. MR 1641677 (2000c:20034), https://doi.org/10.1090/S0002-9939-99-05138-2
  • [8] Anton Evseev, Reduction for characters of finite algebra groups, J. Algebra 325 (2011), 321-351. MR 2745543 (2012a:20024), https://doi.org/10.1016/j.jalgebra.2010.07.048
  • [9] Gustavo A. Fernández-Alcober and Alexander Moretó, On the number of conjugacy class sizes and character degrees in finite $ p$-groups, Proc. Amer. Math. Soc. 129 (2001), no. 11, 3201-3204 (electronic). MR 1844993 (2002d:20009), https://doi.org/10.1090/S0002-9939-01-05946-9
  • [10] Jon González-Sánchez, Kirillov's orbit method for $ p$-groups and pro-$ p$ groups, Comm. Algebra 37 (2009), no. 12, 4476-4488. MR 2588861 (2011a:20057), https://doi.org/10.1080/00927870802545679
  • [11] P. Hall, The classification of prime-power groups, J. Reine Angew. Math. 182 (1940), 130-141. MR 0003389 (2,211b)
  • [12] Marshall Hall Jr., The theory of groups, Chelsea Publishing Co., New York, 1976. Reprinting of the 1968 edition. MR 0414669 (54 #2765)
  • [13] Graham Higman, Enumerating $ p$-groups. I. Inequalities, Proc. London Math. Soc. (3) 10 (1960), 24-30. MR 0113948 (22 #4779)
  • [14] Roger E. Howe, Kirillov theory for compact $ p$-adic groups, Pacific J. Math. 73 (1977), no. 2, 365-381. MR 0579176 (58 #28314)
  • [15] Roger E. Howe, On representations of discrete, finitely generated, torsion-free, nilpotent groups, Pacific J. Math. 73 (1977), no. 2, 281-305. MR 0499004 (58 #16984)
  • [16] Bertram Huppert, A remark on the character-degrees of some $ p$-groups, Arch. Math. (Basel) 59 (1992), no. 4, 313-318. MR 1179454 (93g:20016), https://doi.org/10.1007/BF01197044
  • [17] I. M. Isaacs, Sets of $ p$-powers as irreducible character degrees, Proc. Amer. Math. Soc. 96 (1986), no. 4, 551-552. MR 826479 (87d:20013), https://doi.org/10.2307/2046302
  • [18] I. M. Isaacs, Counting characters of upper triangular groups, J. Algebra 315 (2007), no. 2, 698-719. MR 2351888 (2009a:20023), https://doi.org/10.1016/j.jalgebra.2007.01.027
  • [19] Noboru Ito and Avinoam Mann, Counting classes and characters of groups of prime exponent, Israel J. Math. 156 (2006), 205-220. MR 2282376 (2008i:20010), https://doi.org/10.1007/BF02773832
  • [20] A. Jaikin-Zapirain, Zeta function of representations of compact $ p$-adic analytic groups, J. Amer. Math. Soc. 19 (2006), no. 1, 91-118 (electronic). MR 2169043 (2006f:20029), https://doi.org/10.1090/S0894-0347-05-00501-1
  • [21] Marshall Hall Jr., A basis for free Lie rings and higher commutators in free groups, Proc. Amer. Math. Soc. 1 (1950), 575-581. MR 0038336 (12,388a)
  • [22] Thomas Michael Keller, Derived length and conjugacy class sizes, Adv. Math. 199 (2006), no. 1, 88-103. MR 2187399 (2006k:20027), https://doi.org/10.1016/j.aim.2004.11.002
  • [23] E. I. Khukhro, $ p$-automorphisms of finite $ p$-groups, London Mathematical Society Lecture Note Series, vol. 246, Cambridge University Press, Cambridge, 1998. MR 1615819 (99d:20029)
  • [24] Serge Lang and André Weil, Number of points of varieties in finite fields, Amer. J. Math. 76 (1954), 819-827. MR 0065218 (16,398d)
  • [25] Martin W. Liebeck and Aner Shalev, Character degrees and random walks in finite groups of Lie type, Proc. London Math. Soc. (3) 90 (2005), no. 1, 61-86. MR 2107038 (2006h:20016), https://doi.org/10.1112/S0024611504014935
  • [26] Christophe Reutenauer, Free Lie algebras, London Mathematical Society Monographs. New Series, vol. 7, The Clarendon Press Oxford University Press, New York, 1993. Oxford Science Publications. MR 1231799 (94j:17002)
  • [27] A. Stasinski and C. Voll, Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type $ B$, Amer. J. Math. 136 (2014), no. 2, 501-550. MR 3188068, https://doi.org/10.1353/ajm.2014.0010
  • [28] Ralph Stöhr and Michael Vaughan-Lee, Products of homogeneous subspaces in free Lie algebras, Internat. J. Algebra Comput. 19 (2009), no. 5, 699-703. MR 2547065 (2010i:17011), https://doi.org/10.1142/S0218196709005287
  • [29] A. Vera López and J. M. Arregi, Conjugacy classes in Sylow $ p$-subgroups of $ {\rm GL}(n,q)$. IV, Glasgow Math. J. 36 (1994), no. 1, 91-96. MR 1260823 (94m:20052), https://doi.org/10.1017/S0017089500030597
  • [30] Antonio Vera-López and J. M. Arregi, Conjugacy classes in unitriangular matrices, Linear Algebra Appl. 370 (2003), 85-124. MR 1994321 (2004i:20091), https://doi.org/10.1016/S0024-3795(03)00371-9
  • [31] Christopher Voll, Zeta functions of nilpotent groups--singular Pfaffians, Essays in geometric group theory, Ramanujan Math. Soc. Lect. Notes Ser., vol. 9, Ramanujan Math. Soc., Mysore, 2009, pp. 145-159. MR 2605359 (2011g:20049)

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Additional Information

E. A. O’Brien
Affiliation: Department of Mathematics, University of Auckland, Auckland, New Zealand
Email: obrien@math.auckland.ac.nz

C. Voll
Affiliation: School of Mathematics, University of Southampton, University Road, Southampton SO17 1BJ, United Kingdom
Address at time of publication: Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany
Email: C.Voll.98@cantab.net

DOI: https://doi.org/10.1090/tran/6276
Keywords: Finite $p$-groups, character degrees, conjugacy class sizes, Kirillov orbit methods, Lazard correspondence, relatively free $p$-groups
Received by editor(s): November 28, 2012
Received by editor(s) in revised form: August 7, 2013
Published electronically: April 3, 2015
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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