On the Number of Irreducible Characters and Their Anchor Groups

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DOI: https://doi.org/10.28924/2291-8639-24-2026-148
Published: May 11, 2026
Cite as:
Manal H. Algreagri, On the Number of Irreducible Characters and Their Anchor Groups, Int. J. Anal. Appl., 24 (2026), 148.

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Manal H. Algreagri

Abstract

We prove that for a finite group G, the number of height-zero characters with a fixed anchor subgroup A within a p-block B coincides with the number of height zero characters with anchor A in the Brauer correspondent block of NG(P). The proof proceeds via Brauer’s First Main Theorem. We describe the anchor group of an irreducible character that is contained in the principal p-blocks containing at most five height zero irreducible ordinary characters. We also compute suitable examples.

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References

  1. J. Alperin, the Main Problem of Block Theory, in: Proceedings of the Conference on Finite Groups, Elsevier, 1976: pp. 341–356. https://doi.org/10.1016/B978-0-12-633650-4.50025-4.
  2. M.H. Algreagri, A.M. Alghamdi, Some Remarks on Anchor of Irreducible Characters, Adv. Group Theory Appl. 18 (2024), 107–122. https://doi.org/10.32037/agta-2024-004.
  3. M.H. Algreagri, A.M. Alghamdi, Irreducible Characters with Cyclic Anchor Group, Axioms 12 (2023), 950. https://doi.org/10.3390/axioms12100950.
  4. V.A. Belonogov, Finite Groups With a Small Principal $p$-Block, in: Group-Theoretic Investigations, Akad. Nauk SSSR Ural. Otdel. Sverdlovsk, (1990), 8–30.
  5. J. Brandt, A Lower Bound for the Number of Irreducible Characters in a Block, J. Algebra. 74 (1982), 509–515. https://doi.org/10.1016/0021-8693(82)90036-9.
  6. R. Brauer, On the Arithmetic in a Group Ring, Proc. Natl. Acad. Sci. USA 30 (1944), 109–114. https://doi.org/10.1073/pnas.30.5.109.
  7. R. Brauer, Representations of Finite Groups, in: Lectures on Modern Mathematics, Vol. 1, Wiley, 1963.
  8. R. Brauer, On Blocks of Characters of Groups of Finite Order, Proc. Natl. Acad. Sci. USA 32 (1946), 182–186. https://doi.org/10.1073/pnas.32.6.182.
  9. R. Brauer, Notes on Representations of Finite Groups, I, J. Lond. Math. Soc. s2-13 (1976), 162–166. https://doi.org/10.1112/jlms/s2-13.1.162.
  10. C. W. Curtis, I. Reiner, Methods of Representation Theory–With Applications to Finite Groups and Orders, Wiley, New York, 1981.
  11. C.W. Curtis, I. Reiner, Methods of Representation Theory, Wiley-Interscience, New York, 1981.
  12. D.A. Craven, Representation Theory of Finite Groups: A Guidebook, Springer, 2019. https://doi.org/10.1007/978-3-030-21792-1.
  13. E.C. Dade, A Correspondence of Characters, in: The Santa Cruz Conference on Finite Groups, Proc. Sympos. Pure Math. vol. 37, Univ. California, Santa Cruz, CA, 1979, American Mathematical Society, Providence, pp. 401–403, (1980).
  14. W. Feit, The Representation Theory of Finite Groups, North-Holland Mathematical Library, Vol. 25, North-Holland, Amsterdam, (1982).
  15. J.A. Green, Some Remarks on Defect Groups, Math. Z. 107 (1968), 133–150. https://doi.org/10.1007/BF01111026.
  16. E. Giannelli, N. Rizo, B. Sambale, A. Schaeffer Fry, Groups with Few $acute{p}$-Character Degrees in the Principal Block, Proc. Am. Math. Soc. 148 (2020), 4597–4614. https://doi.org/10.1090/proc/15143.
  17. N.N. Hung, A. Schaeffer Fry, C. Vallejo, Height Zero Characters in Principal Blocks, J. Algebra. 622 (2023), 197–219. https://doi.org/10.1016/j.jalgebra.2023.01.021.
  18. N.H. Nguyen, A.A. Schaeffer Fry, On Héthelyi–Külshammer’s Conjecture for Principal Blocks, Algebr. Number Theory 17 (2023), 1127–1151. https://doi.org/10.2140/ant.2023.17.1127.
  19. R. Kessar, B. Külshammer, M. Linckelmann, Anchors of Irreducible Characters, J. Algebra. 475 (2017), 113–132. https://doi.org/10.1016/j.jalgebra.2015.11.034.
  20. S. Koshitani, T. Sakurai, The Principal $p$‐Blocks with Four Irreducible Characters, Bull. Lond. Math. Soc. 53 (2021), 1124–1138. https://doi.org/10.1112/blms.12488.
  21. P. Landrock, On the Number of Irreducible Characters in a 2-Block, J. Algebra. 68 (1981), 426–442. https://doi.org/10.1016/0021-8693(81)90272-6.
  22. M. Linckelmann, The Block Theory of Finite Group Algebras: Volume 1, Cambridge University Press, Cambridge, 2018.
  23. M. Linckelmann, The Block Theory of Finite Group Algebras: Volume 2, Cambridge University Press, Cambridge, 2018.
  24. H. Nagao, Y. Tsushima, Representations of Finite Groups, Academic Press, (1989).
  25. G. Navarro, Characters and Blocks of Finite Groups, Cambridge University Press, Cambridge, 1998.
  26. G. Navarro, B. Sambale, P.H. Tiep, Characters and Sylow 2-Subgroups of Maximal Class Revisited, J. Pure Appl. Algebr. 222 (2018), 3721–3732. https://doi.org/10.1016/j.jpaa.2018.02.002.
  27. T. Okuyama, and M. Wajima, Character Correspondence and $p$-Blocks of $P$-Solvable Groups, Osaka J. Math. 17 (1980), 801–806.
  28. N. Rizo, A. Schaeffer Fry, C. Vallejo, Principal Blocks with 5 Irreducible Characters, J. Algebra. 585 (2021), 316–337. https://doi.org/10.1016/j.jalgebra.2021.06.009.
  29. The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.12.1, 2022, https://www.gap-system.org.
  30. J. Thévenaz, G-Algebras and Modular Representation Theory (Oxford Mathematical Monographs), Oxford University Press, 1995.
  31. A.V. López, J.V. López, Classification of Finite Groups According to the Number of Conjugacy Classes, Isr. J. Math. 51 (1985), 305–338. https://doi.org/10.1007/bf02764723.
  32. P. Webb, A Course in Finite Group Representation Theory, Cambridge University Press, 2016.