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On almost p-rational characters of p ′ -degree

  • Nguyen Ngoc Hung EMAIL logo , Gunter Malle and Attila Maróti
Published/Copyright: October 26, 2022
Forum Mathematicum
From the journal Forum Mathematicum Volume 34 Issue 6

Abstract

Let p be a prime and let G be a finite group. A complex character of G is called almost p-rational if its values belong to a cyclotomic field ℚ ⁢ ( e 2 ⁢ π ⁢ i / n ) for some n ∈ ℤ + not divisible by p 2 . We prove that, in contrast to usual p-rational characters, there are “many” almost p-rational irreducible characters in finite groups. We obtain both explicit and asymptotic bounds for the number of almost p-rational irreducible characters of G in terms of p. In fact, motivated by the McKay–Navarro conjecture, we obtain the same bound for the number of such characters of p ′ -degree and prove that, in the minimal situation, the number of almost p-rational irreducible p ′ -characters of G coincides with that of 𝐍 G ⁢ ( P ) for P ∈ Syl p ⁡ ( G ) . Lastly, we propose a new way to detect the cyclicity of Sylow p-subgroups of a finite group G from its character table, using almost p-rational irreducible p ′ -characters and the blockwise refinement of the McKay–Navarro conjecture.

Keywords: Finite groups; McKay–Navarro conjecture; linear actions; coprime actions
MSC 2010: 20C15; 20E45; 20D20; 20D05; 20D10

Communicated by Manfred Droste

Funding source: Deutsche Forschungsgemeinschaft

Award Identifier / Grant number: 286237555

Funding source: H2020 European Research Council

Award Identifier / Grant number: 741420

Funding source: Nemzeti Kutatási Fejlesztési és Innovációs Hivatal

Award Identifier / Grant number: K132951

Award Identifier / Grant number: K138596

Award Identifier / Grant number: K138828

Funding statement: The second author gratefully acknowledges support by the DFG, TRR 195, Project-ID 286237555. The work of the third author on the project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 741420), and is supported by the Hungarian National Research, Development and Innovation Office (NKFIH) Grant nos. K132951, K138596 and K138828.

Acknowledgements

We thank Gabriel Navarro for his insightful comments on an earlier version of the manuscript.

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Received: 2022-01-04
Revised: 2022-09-06
Published Online: 2022-10-26
Published in Print: 2022-11-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

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  2. Wong–Zakai approximations and support theorems for stochastic McKean–Vlasov equations
  3. The Singer transfer for infinite real projective space
  4. The Stiefel–Whitney classes of moment-angle manifolds are trivial
  5. On almost p-rational characters of p ′ -degree
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https://doi.org/10.1515/forum-2022-0005
Keywords for this article
Finite groups; McKay–Navarro conjecture; linear actions; coprime actions

Articles in the same Issue

  1. Frontmatter
  2. Wong–Zakai approximations and support theorems for stochastic McKean–Vlasov equations
  3. The Singer transfer for infinite real projective space
  4. The Stiefel–Whitney classes of moment-angle manifolds are trivial
  5. On almost p-rational characters of p ′ -degree
  6. Homogeneous four-manifolds with half-harmonic Weyl curvature tensor
  7. Construction of a class of spectral measures
  8. On the number of prime divisors and radicals of non-zero Fourier coefficients of Hilbert cusp forms
  9. Molecular characterization of weak Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type with its applications to Littlewood–Paley function characterizations
  10. Conway invariant Jacobi forms on the Leech lattice
  11. Geometric characterizations of Gromov hyperbolic Hölder domains
  12. Improved quantitative unique continuation for complex-valued drift equations in the plane
  13. On uniqueness of branching to fixed point Lie subalgebras
  14. Fefferman type criterion on weighted bi-parameter local Hardy spaces and boundedness of bi-parameter pseudodifferential operators
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