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Character triples and relative defect zero characters

  • Junwei Zhang , Lizhong Wang und Ping Jin EMAIL logo
Veröffentlicht/Copyright: 27. Februar 2025
Journal of Group Theory
Aus der Zeitschrift Journal of Group Theory Band 28 Heft 4

Abstract

Given a character triple ( G , N , θ ) , which means that 𝐺 is a finite group with N ⁢ ⊲ ⁢ G and θ ∈ Irr ⁡ ( N ) is 𝐺-invariant, we introduce the notion of a 𝜋-quasi-extension of 𝜃 to 𝐺, where 𝜋 is the set of primes dividing the order of the cohomology element [ θ ] G / N ∈ H 2 ⁢ ( G / N , C × ) associated with the character triple, and then we establish the uniqueness of such an extension in the normalized case. As an application, we use the 𝜋-quasi-extension of 𝜃 to construct a bijection from the set of 𝜋-defect zero characters of G / N onto the set of relative 𝜋-defect zero characters of 𝐺 over 𝜃. Our results generalize the related theorems of M. Murai and of G. Navarro.

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12431001

Award Identifier / Grant number: 12171289

Funding statement: This work was supported by the NSF of China (Nos. 12431001, 12171289).

Acknowledgements

The authors are grateful to the referee for valuable comments and suggestions that greatly improve the exposition of the paper, and to Professor Jiping Zhang for his guidance and encouragement.

  1. Communicated by: Hung Tong-Viet

References

[1] K. S. Brown, Cohomology of Groups, Grad. Texts in Math. 87, Springer, New York, 1982. 10.1007/978-1-4684-9327-6Suche in Google Scholar

[2] J. P. Cossey and M. L. Lewis, Lifts and vertex pairs in solvable groups, Proc. Edinb. Math. Soc. (2) 55 (2012), no. 1, 143–153. 10.1017/S001309151000129XSuche in Google Scholar

[3] I. M. Isaacs, Character Theory of Finite Groups, Pure Appl. Math. 69, Academic Press, New York, 1976. Suche in Google Scholar

[4] I. M. Isaacs and G. Navarro, Weights and vertices for characters of 𝜋-separable groups, J. Algebra 177 (1995), no. 2, 339–366. 10.1006/jabr.1995.1301Suche in Google Scholar

[5] M. Murai, Defect zero characters and relative defect zero characters, Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), no. 9, 149–151. 10.3792/pjaa.88.149Suche in Google Scholar

[6] G. Navarro, Weights, vertices and a correspondence of characters in groups of odd order, Math. Z. 212 (1993), no. 4, 535–544. 10.1007/BF02571674Suche in Google Scholar

[7] G. Navarro, Brauer characters relative to a normal subgroup, Proc. Lond. Math. Soc. (3) 81 (2000), no. 1, 55–71. 10.1112/S0024611500012351Suche in Google Scholar

[8] G. Navarro, Defect of characters and local subgroups in 𝑝-solvable groups, J. Group Theory 5 (2002), no. 3, 241–268. 10.1515/jgth.2002.001Suche in Google Scholar

[9] G. Navarro, Vertices for characters of 𝑝-solvable groups, Trans. Amer. Math. Soc. 354 (2002), no. 7, 2759–2773. 10.1090/S0002-9947-02-02974-4Suche in Google Scholar

[10] G. Navarro, A new character correspondence in groups of odd order, J. Algebra 268 (2003), no. 1, 8–21. 10.1016/S0021-8693(03)00420-4Suche in Google Scholar

[11] G. Navarro, Actions and characters in blocks, J. Algebra 275 (2004), no. 2, 471–480. 10.1016/j.jalgebra.2004.01.022Suche in Google Scholar

[12] G. Navarro, Character Theory and the McKay Conjecture, Cambridge Stud. Adv. Math. 175, Cambridge University, Cambridge, 2018. 10.1017/9781108552790Suche in Google Scholar

[13] G. Navarro and B. Sambale, Weights and nilpotent subgroups, Int. Math. Res. Not. IMRN 2021 (2021), no. 4, 2526–2538. 10.1093/imrn/rnz195Suche in Google Scholar

[14] G. Navarro and P. H. Tiep, A reduction theorem for the Alperin weight conjecture, Invent. Math. 184 (2011), no. 3, 529–565. 10.1007/s00222-010-0295-2Suche in Google Scholar

[15] P. Schmid, Extending irreducible representations of normal subgroups, J. Algebra 94 (1985), no. 1, 30–51. 10.1016/0021-8693(85)90203-0Suche in Google Scholar
Received: 2024-08-23
Revised: 2025-01-18
Published Online: 2025-02-27
Published in Print: 2025-07-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. On the kernel of actions on asymptotic cones
  3. CAT(0) cube complexes and asymptotically rigid mapping class groups
  4. Iwip endomorphisms of free groups and fixed points of graph selfmaps
  5. Space of orders with finite Cantor–Bendixson rank
  6. Lifting subgroups of PSL2 to SL2 over local fields
  7. Regular 3-polytopes of order 2𝑛𝑝
  8. On the number of tuples of group elements satisfying a first-order formula
  9. Exponent-critical groups
  10. On soluble groups in which commutators have prime power order
  11. A character theoretic criterion for Fitting height
  12. Hilbert divisors and degrees of irreducible Brauer characters
  13. Character triples and relative defect zero characters
https://doi.org/10.1515/jgth-2024-0176

Artikel in diesem Heft

  1. Frontmatter
  2. On the kernel of actions on asymptotic cones
  3. CAT(0) cube complexes and asymptotically rigid mapping class groups
  4. Iwip endomorphisms of free groups and fixed points of graph selfmaps
  5. Space of orders with finite Cantor–Bendixson rank
  6. Lifting subgroups of PSL2 to SL2 over local fields
  7. Regular 3-polytopes of order 2𝑛𝑝
  8. On the number of tuples of group elements satisfying a first-order formula
  9. Exponent-critical groups
  10. On soluble groups in which commutators have prime power order
  11. A character theoretic criterion for Fitting height
  12. Hilbert divisors and degrees of irreducible Brauer characters
  13. Character triples and relative defect zero characters
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