Algebra & Number Theory

Groups with exactly one irreducible character of degree divisible by $p$

Daniel Goldstein, Robert Guralnick, Mark Lewis, Alexander Moretó, Gabriel Navarro, and Huu Tiep Pham

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Let p be a prime. We characterize those finite groups which have precisely one irreducible character of degree divisible by p.

Article information

Algebra Number Theory, Volume 8, Number 2 (2014), 397-428.

Received: 20 February 2013
Revised: 6 May 2013
Accepted: 11 July 2013
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20C15: Ordinary representations and characters

finite groups characters character degrees


Goldstein, Daniel; Guralnick, Robert; Lewis, Mark; Moretó, Alexander; Navarro, Gabriel; Pham, Huu Tiep. Groups with exactly one irreducible character of degree divisible by $p$. Algebra Number Theory 8 (2014), no. 2, 397--428. doi:10.2140/ant.2014.8.397.

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  • C. Casolo and S. Dolfi, “Products of primes in conjugacy class sizes and irreducible character degrees”, Israel J. Math. 174 (2009), 403–418.
  • D. Chillag and I. D. Macdonald, “Generalized Frobenius groups”, Israel J. Math. 47:2-3 (1984), 111–122.
  • S. Dolfi, A. Moretó, and G. Navarro, “The groups with exactly one class of size a multiple of $p$”, J. Group Theory 12:2 (2009), 219–234.
  • S. Dolfi, G. Navarro, and P. H. Tiep, “Finite groups whose same degree characters are Galois conjugate”, Israel J. Math. 198:1 (2013), 283–331.
  • W. Feit, “Extending Steinberg characters”, pp. 1–9 in Linear algebraic groups and their representations (Los Angeles, 1992), edited by R. S. Elman et al., Contemp. Math. 153, Amer. Math. Soc., Providence, RI, 1993.
  • G. A. Fernández-Alcober and A. Moretó, “Groups with two extreme character degrees and their normal subgroups”, Trans. Amer. Math. Soc. 353:6 (2001), 2171–2192.
  • S. M. Gagola, Jr., “Characters vanishing on all but two conjugacy classes”, Pacific J. Math. 109:2 (1983), 363–385.
  • The GAP group, GAP,–-,groups, algorithms, and programming, version 4.4, 2004,
  • R. M. Guralnick and J. Saxl, “Generation of finite almost simple groups by conjugates”, J. Algebra 268:2 (2003), 519–571.
  • R. M. Guralnick and P. H. Tiep, “The non-coprime $k(GV)$ problem”, J. Algebra 293:1 (2005), 185–242.
  • B. Huppert, “Zweifach transitive, auflösbare Permutationsgruppen”, Math. Z. 68 (1957), 126–150.
  • B. Huppert and N. Blackburn, Finite groups, III, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 243, Springer, Berlin, 1982.
  • I. M. Isaacs, Character theory of finite groups, Pure and Applied Mathematics 69, Academic Press, New York, 1976.
  • I. M. Isaacs, “Coprime group actions fixing all nonlinear irreducible characters”, Canad. J. Math. 41:1 (1989), 68–82.
  • I. M. Isaacs, “Bounding the order of a group with a large character degree”, J. Algebra 348 (2011), 264–275.
  • I. M. Isaacs, A. Moretó, G. Navarro, and P. H. Tiep, “Groups with just one character degree divisible by a given prime”, Trans. Amer. Math. Soc. 361:12 (2009), 6521–6547.
  • C. Jansen, K. Lux, R. Parker, and R. Wilson, An atlas of Brauer characters, London Mathematical Society Monographs. New Series 11, Oxford University Press, New York, 1995.
  • A. I. Kostrikin and P. H. Tiep, Orthogonal decompositions and integral lattices, Expositions in Mathematics 15, de Gruyter, Berlin, 1994.
  • M. L. Lewis, “Bounding Fitting heights of character degree graphs”, J. Algebra 242:2 (2001), 810–818.
  • M. L. Lewis and D. L. White, “Connectedness of degree graphs of nonsolvable groups”, J. Algebra 266:1 (2003), 51–76.
  • M. L. Lewis, A. Moretó, and T. R. Wolf, “Non-divisibility among character degrees”, J. Group Theory 8:5 (2005), 561–588.
  • M. W. Liebeck, “The affine permutation groups of rank three”, Proc. London Math. Soc. $(3)$ 54:3 (1987), 477–516.
  • O. Manz and T. R. Wolf, Representations of solvable groups, London Mathematical Society Lecture Note Series 185, Cambridge University Press, 1993.
  • O. Manz, R. Staszewski, and W. Willems, “On the number of components of a graph related to character degrees”, Proc. Amer. Math. Soc. 103:1 (1988), 31–37.
  • J. McKay, “Graphs, singularities, and finite groups”, pp. 183–186 in The Santa Cruz Conference on Finite Groups (Santa Cruz, CA, 1979), edited by B. Cooperstein and G. Mason, Proc. Sympos. Pure Math. 37, Amer. Math. Soc., Providence, R.I., 1980.
  • G. Navarro and P. H. Tiep, “Rational irreducible characters and rational conjugacy classes in finite groups”, Trans. Amer. Math. Soc. 360:5 (2008), 2443–2465.
  • T. Noritzsch, “Groups having three complex irreducible character degrees”, J. Algebra 175:3 (1995), 767–798.
  • D. Passman, Permutation groups, W. A. Benjamin, New York and Amsterdam, 1968.
  • C. H. Sah, “Cohomology of split group extensions, II”, J. Algebra 45:1 (1977), 17–68.
  • G. Seitz, “Finite groups having only one irreducible representation of degree greater than one”, Proc. Amer. Math. Soc. 19 (1968), 459–461.
  • R. Steinberg, “Finite subgroups of ${\rm SU}\sb 2$, Dynkin diagrams and affine Coxeter elements”, Pacific J. Math. 118:2 (1985), 587–598.