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Character degrees of 5-groups of maximal class

  • Lijuan He , Heng Lv EMAIL logo and Dongfang Yang
Published/Copyright: March 6, 2024
Journal of Group Theory
From the journal Journal of Group Theory Volume 27 Issue 5

Abstract

Let 𝐺 be a 5-group of maximal class with major centralizer G 1 = C G ⁒ ( G 2 / G 4 ) . In this paper, we prove that the irreducible character degrees of a 5-group 𝐺 of maximal class are almost determined by the irreducible character degrees of the major centralizer G 1 and show that the set of irreducible character degrees of a 5-group of maximal class is either { 1 , 5 , 5 3 } or { 1 , 5 , … , 5 k } with k β‰₯ 1 .

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11971391

Award Identifier / Grant number: 12071376

Award Identifier / Grant number: 12301018

Funding source: Fundamental Research Funds for the Central Universities

Award Identifier / Grant number: SWU-XDJH202305

Funding source: Natural Science Foundation of Jiangsu Province

Award Identifier / Grant number: 23KJB110002

Funding statement: This research is supported by the National Natural Science Foundation of China (Nos. 11971391, 12071376) and Fundamental Research Funds for the Central Universities (SWU-XDJH202305). The third author is supported by the NSF of China (No. 12301018) and the Natural Science Foundation for the Universities in Jiangsu Province (No. 23KJB110002).

Acknowledgements

The authors would like to thank the referee for her or his valuable suggestions and useful comments on this paper. They particularly thank the referee for the statement and proof of Lemma 3.10, which have greatly improved the quality of this paper.

  1. Communicated by: Hung Tong-Viet

References

[1] Y. Berkovich, Groups of Prime Power Order. Vol. 1, De Gruyter Exp. Math. 46, Walter de Gruyter, Berlin, 2008. 10.1515/9783110208221Search in Google Scholar

[2] T. Bonner and L. Wilson, On the Taketa bound for normally monomial 𝑝-groups of maximal class II, J. Algebra Appl. 13 (2014), no. 5, Article ID 1350145. 10.1142/S0219498813501454Search in Google Scholar

[3] W. Brisley and I. D. Macdonald, Two classes of metabelian 𝑝-groups, Math. Z. 112 (1969), 5–12. 10.1007/BF01277489Search in Google Scholar

[4] B. Huppert, Endliche Gruppen. I, Grundlehren Math. Wiss. 134, Springer, Berlin, 1967. 10.1007/978-3-642-64981-3Search in Google Scholar

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

[6] T. M. Keller, D. Ragan and G. T. Tims, On the Taketa bound for normally monomial 𝑝-groups of maximal class, J. Algebra 277 (2004), no. 2, 657–688. 10.1016/j.jalgebra.2003.11.018Search in Google Scholar

[7] L. G. KovΓ‘cs and C. R. Leedham-Green, Some normally monomial 𝑝-groups of maximal class and large derived length, Quart. J. Math. Oxford Ser. (2) 37 (1986), no. 145, 49–54. 10.1093/qmath/37.1.49Search in Google Scholar

[8] C. R. Leedham-Green and S. McKay, The Structure of Groups of Prime Power Order, London Math. Soc. Monogr. (N. S.) 27, Oxford University, Oxford, 2002. 10.1093/oso/9780198535485.001.0001Search in Google Scholar

[9] A. Mann, Normally monomial 𝑝-groups, J. Algebra 300 (2006), no. 1, 2–9. 10.1016/j.jalgebra.2005.06.027Search in Google Scholar

[10] A. Mann, Character degrees of some 𝑝-groups, preprint (2016), https://arxiv.org/abs/1602.04689. Search in Google Scholar

[11] M. C. Slattery, Character degrees of normally monomial maximal class 5-groups, Character Theory of Finite Groups, Contemp. Math. 524, American Mathematical Society, Providence (2010), 153–159. 10.1090/conm/524/10354Search in Google Scholar

[12] M. C. Slattery, Maximal class 𝑝-groups with large character degree gaps, Arch. Math. (Basel) 105 (2015), no. 6, 501–507. 10.1007/s00013-015-0836-4Search in Google Scholar

[13] D. Yang and H. Lv, Character degrees of normally monomial 𝑝-groups of maximal class, J. Group Theory 26 (2023), no. 4, 817–826. 10.1515/jgth-2021-0212Search in Google Scholar
Received: 2023-07-09
Revised: 2024-02-06
Published Online: 2024-03-06
Published in Print: 2024-09-01

Β© 2024 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. The relational complexity of linear groups acting on subspaces
  3. Cliques in derangement graphs for innately transitive groups
  4. Representation zeta function of a family of maximal class groups: Non-exceptional primes
  5. Character degrees of 5-groups of maximal class
  6. Automorphic word maps and the Amit–Ashurst conjecture
  7. Groups with subnormal or modular Schmidt 𝑝𝑑-subgroups
  8. Finite normal subgroups of strongly verbally closed groups
  9. The central tree property and algorithmic problems on subgroups of free groups
  10. Uniqueness of roots up to conjugacy in circular and hosohedral-type Garside groups
  11. Isomorphisms and commensurability of surface Houghton groups
https://doi.org/10.1515/jgth-2023-0103

Articles in the same Issue

  1. Frontmatter
  2. The relational complexity of linear groups acting on subspaces
  3. Cliques in derangement graphs for innately transitive groups
  4. Representation zeta function of a family of maximal class groups: Non-exceptional primes
  5. Character degrees of 5-groups of maximal class
  6. Automorphic word maps and the Amit–Ashurst conjecture
  7. Groups with subnormal or modular Schmidt 𝑝𝑑-subgroups
  8. Finite normal subgroups of strongly verbally closed groups
  9. The central tree property and algorithmic problems on subgroups of free groups
  10. Uniqueness of roots up to conjugacy in circular and hosohedral-type Garside groups
  11. Isomorphisms and commensurability of surface Houghton groups
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