Illinois Journal of Mathematics

Extending Huppert’s conjecture from non-Abelian simple groups to quasi-simple groups

Nguyen Ngoc Hung, Philani R. Majozi, Hung P. Tong-Viet, and Thomas P. Wakefield

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We propose to extend a conjecture of Bertram Huppert [Illinois J. Math. 44 (2000) 828–842] from finite non-Abelian simple groups to finite quasi-simple groups. Specifically, we conjecture that if a finite group $G$ and a finite quasi-simple group $H$ with ${\mathrm{Mult}}(H/\mathbf{Z}(H))$ cyclic have the same set of irreducible character degrees (not counting multiplicity), then $G$ is isomorphic to a central product of $H$ and an Abelian group. We present a pattern to approach this extended conjecture and, as a demonstration, we confirm it for the special linear groups in dimensions $2$ and $3$.

Article information

Illinois J. Math., Volume 59, Number 4 (2015), 901-924.

Received: 9 November 2016
First available in Project Euclid: 27 February 2017

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Zentralblatt MATH identifier

Primary: 20C15: Ordinary representations and characters
Secondary: 20C33: Representations of finite groups of Lie type 20C34: Representations of sporadic groups 20C30: Representations of finite symmetric groups


Hung, Nguyen Ngoc; Majozi, Philani R.; Tong-Viet, Hung P.; Wakefield, Thomas P. Extending Huppert’s conjecture from non-Abelian simple groups to quasi-simple groups. Illinois J. Math. 59 (2015), no. 4, 901--924. doi:10.1215/ijm/1488186014.

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  • C. Bessenrodt, H. N. Nguyen, J. B. Olsson and H. P. Tong-Viet, Complex group algebras of the double covers of the symmetric and alternating groups, Algebra Number Theory 9 (2015), 601–628.
  • M. Bianchi, D. Chillag, M. L. Lewis and E. Pacifici, Character degree graphs that are complete graphs, Proc. Amer. Math. Soc. 135 (2007), 671–676.
  • T. Breuer, Manual for the GAP character table library, Version 1.1, RWTH Aachen, 2004.
  • R. W. Carter, Finite groups of Lie type. Conjugacy classes and complex characters, Wiley, New York, 1985.
  • J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of finite groups, Oxford University Press, London, 1984.
  • J. S. Frame and W. A. Simpson, The character tables for $\mathrm{SL}_3(q)$, $\mathrm {SU}_3(q)$, $\mathrm{PSL}_3(q)$, $\mathrm{PSU}_3(q^2)$, Canad. J. Math. 25 (1973), 486–494.
  • D. Gorenstein, R. Lyons and R. Solomon, The classification of the finite simple groups, Number 3, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1998.
  • B. Huppert, Some simple groups which are determined by the set of their character degrees I, Illinois J. Math. 44 (2000), 828–842.
  • I. M. Isaacs, Character theory of finite groups, Dover Publications, New York, 1994.
  • I. M. Isaacs, G. Malle and G. Navarro, A reduction theorem for the McKay conjecture, Invent. Math. 170 (2007), 33–101.
  • O. H. King, The subgroup structure of finite classical groups in terms of geometric configurations, London. Math. Soc. Lecture Note Series 327 (2005), 29–56.
  • M. Lewis and D. White, Connectedness of degree graphs of non-solvable groups, J. Algebra 266 (2003), 51–76.
  • G. Malle, Extensions of unipotent characters and the inductive McKay condition, J. Algebra 320 (2008), 2963–2980.
  • A. Moretó, An answer to a question of Isaacs on character degree graphs, Adv. Math. 201 (2006), 90–101.
  • A. Moretó, Complex group algebras of finite groups: Brauer's problem 1, Adv. Math. 208 (2007), 236–248.
  • T. Nagell, Des équations indéterminées $x^2+x+1=y^n$ et $x^2+x+1=3y^n$, Norsk. mat. Foren. Skr. Serie I (1921), 1–14.
  • G. Navarro, The set of character degrees of a finite group does not determine its solvability, Proc. Amer. Math. Soc. 143 (2015), 989–990.
  • G. Navarro and N. Rizo, Nilpotent and perfect groups with the same set of character degrees, J. Algebra Appl. 13 (2014), 1450061. 3 pp.
  • G. Navarro, P. H. Tiep and A. Turull, Brauer characters with cyclotomic field of values, J. Pure Appl. Algebra 212 (2008), 628–635.
  • H. N. Nguyen, Quasisimple classical groups and their complex group algebras, Israel J. Math. 195 (2013), 973–998.
  • H. N. Nguyen and H. P. Tong-Viet, Characterizing finite quasisimple groups by their complex group algebras, Algebr. Represent. Theory 17 (2014), 305–320.
  • H. N. Nguyen and H. P. Tong-Viet, Recognition of finite quasi-simple groups by the degrees of their irreducible representations, Groups St Andrews 2013, London Math. Soc. Lecture Note Ser., vol. 422, Cambridge Univ. Press, Cambridge, 2015, pp. 439–456.
  • H. N. Nguyen, H. P. Tong-Viet and T. P. Wakefield, Projective special linear groups $\mathrm{PSL}_4(q)$ are determined by the set of their character degrees, J. Algebra Appl. 11 (2012), 1250108. 26 pp.
  • H. P. Tong-Viet, The simple Ree groups ${}^2\mathrm{F}_4(q^2)$ are determined by the set of the character degrees, J. Algebra 339 (2011), 357–369.
  • H. P. Tong-Viet and T. P. Wakefield, On Huppert's conjecture for $\mathrm{G}_2(q)$, $q \geq7$, J. Pure Appl. Algebra 216 (2012), 2720–2729.
  • H. P. Tong-Viet and T. P. Wakefield, On Huppert's conjecture for ${}^3\mathrm{D}_4(q)$, $q\geq3$, Algebr. Represent. Theory 16 (2013), 470–490.
  • T. P. Wakefield, Verifying Huppert's conjecture for the simple groups of Lie type of rank two, Ph.D. thesis, Kent State University, 2008.
  • T. P. Wakefield, Verifying Huppert's conjecture for $\mathrm{PSL}_3(q)$ and $\mathrm{PSU}_3(q^2)$, Comm. Algebra 37 (2009), 2887–2906.
  • D. L. White, Character degrees of extensions of ${\mathrm{PSL}}_2(q)$ and ${\mathrm{SL}}_2(q)$, J. Group Theory 16 (2013), 1–33.
  • K. Zsigmondy, Zur theorie der Potenzreste, Monath. Math. Phys. 3 (1892), 265–284.