Enumerating classes and characters of $p$-groups
HTML articles powered by AMS MathViewer
- by E. A. O’Brien and C. Voll PDF
- Trans. Amer. Math. Soc. 367 (2015), 7775-7796 Request permission
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.References
- 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, DOI 10.1215/00127094-1959198
- Edward A. Bender, On Buckheister’s enumeration of $n$ $\times \ n$ matrices, J. Combinatorial Theory Ser. A 17 (1974), 273–274. MR 347622, DOI 10.1016/0097-3165(74)90021-1
- 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, DOI 10.1016/j.jalgebra.2004.03.006
- Mitya Boyarchenko, Representations of unipotent groups over local fields and Gutkin’s conjecture, Math. Res. Lett. 18 (2011), no. 3, 539–557. MR 2802587, DOI 10.4310/MRL.2011.v18.n3.a14
- 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, DOI 10.1007/s11856-008-1004-3
- 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 75983
- 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, DOI 10.1090/S0002-9939-99-05138-2
- Anton Evseev, Reduction for characters of finite algebra groups, J. Algebra 325 (2011), 321–351. MR 2745543, DOI 10.1016/j.jalgebra.2010.07.048
- 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. MR 1844993, DOI 10.1090/S0002-9939-01-05946-9
- Jon González-Sánchez, Kirillov’s orbit method for $p$-groups and pro-$p$ groups, Comm. Algebra 37 (2009), no. 12, 4476–4488. MR 2588861, DOI 10.1080/00927870802545679
- P. Hall, The classification of prime-power groups, J. Reine Angew. Math. 182 (1940), 130–141. MR 3389, DOI 10.1515/crll.1940.182.130
- Marshall Hall Jr., The theory of groups, Chelsea Publishing Co., New York, 1976. Reprinting of the 1968 edition. MR 0414669
- Graham Higman, Enumerating $p$-groups. I. Inequalities, Proc. London Math. Soc. (3) 10 (1960), 24–30. MR 113948, DOI 10.1112/plms/s3-10.1.24
- Roger E. Howe, Kirillov theory for compact $p$-adic groups, Pacific J. Math. 73 (1977), no. 2, 365–381. MR 579176, DOI 10.2140/pjm.1977.73.365
- Roger E. Howe, On representations of discrete, finitely generated, torsion-free, nilpotent groups, Pacific J. Math. 73 (1977), no. 2, 281–305. MR 499004, DOI 10.2140/pjm.1977.73.281
- Bertram Huppert, A remark on the character-degrees of some $p$-groups, Arch. Math. (Basel) 59 (1992), no. 4, 313–318. MR 1179454, DOI 10.1007/BF01197044
- I. M. Isaacs, Sets of $p$-powers as irreducible character degrees, Proc. Amer. Math. Soc. 96 (1986), no. 4, 551–552. MR 826479, DOI 10.1090/S0002-9939-1986-0826479-1
- I. M. Isaacs, Counting characters of upper triangular groups, J. Algebra 315 (2007), no. 2, 698–719. MR 2351888, DOI 10.1016/j.jalgebra.2007.01.027
- Noboru Ito and Avinoam Mann, Counting classes and characters of groups of prime exponent, Israel J. Math. 156 (2006), 205–220. MR 2282376, DOI 10.1007/BF02773832
- A. Jaikin-Zapirain, Zeta function of representations of compact $p$-adic analytic groups, J. Amer. Math. Soc. 19 (2006), no. 1, 91–118. MR 2169043, DOI 10.1090/S0894-0347-05-00501-1
- Marshall Hall Jr., A basis for free Lie rings and higher commutators in free groups, Proc. Amer. Math. Soc. 1 (1950), 575–581. MR 38336, DOI 10.1090/S0002-9939-1950-0038336-7
- Thomas Michael Keller, Derived length and conjugacy class sizes, Adv. Math. 199 (2006), no. 1, 88–103. MR 2187399, DOI 10.1016/j.aim.2004.11.002
- E. I. Khukhro, $p$-automorphisms of finite $p$-groups, London Mathematical Society Lecture Note Series, vol. 246, Cambridge University Press, Cambridge, 1998. MR 1615819, DOI 10.1017/CBO9780511526008
- Serge Lang and André Weil, Number of points of varieties in finite fields, Amer. J. Math. 76 (1954), 819–827. MR 65218, DOI 10.2307/2372655
- 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, DOI 10.1112/S0024611504014935
- 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
- 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, DOI 10.1353/ajm.2014.0010
- 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, DOI 10.1142/S0218196709005287
- A. Vera López and J. M. Arregi, Conjugacy classes in Sylow $p$-subgroups of $\textrm {GL}(n,q)$. IV, Glasgow Math. J. 36 (1994), no. 1, 91–96. MR 1260823, DOI 10.1017/S0017089500030597
- Antonio Vera-López and J. M. Arregi, Conjugacy classes in unitriangular matrices, Linear Algebra Appl. 370 (2003), 85–124. MR 1994321, DOI 10.1016/S0024-3795(03)00371-9
- 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
Additional Information
- E. A. O’Brien
- Affiliation: Department of Mathematics, University of Auckland, Auckland, New Zealand
- MR Author ID: 251889
- 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
- Received by editor(s): November 28, 2012
- Received by editor(s) in revised form: August 7, 2013
- Published electronically: April 3, 2015
- © Copyright 2015
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 367 (2015), 7775-7796
- MSC (2010): Primary 20C15, 20D15
- DOI: https://doi.org/10.1090/tran/6276
- MathSciNet review: 3391899