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A Study On the Kernels of Irreducible Characters of Finite Groups

Year 2022, Volume 43, Issue 3, 449 - 453, 30.09.2022
https://doi.org/10.17776/csj.904864

Abstract

Let G be a finite group and χ∈Irr(G), where Irr(G) denotes the set of all irreducible characters of G. The kernel of χ is defined by ker⁡(χ)={ g∈G ┤| χ(g)=χ(1)}, where χ(1) is the character degree of χ. The irreducible character χ of G is called as monolithic when the factor group G/ker⁡(χ) has only one minimal normal subgroup. In this study, we have proven some results by concentrating on the kernels of nonlinear irreducible characters of G. First, we have provided an alternative proof for the classification of finite groups possessing two nonlinear irreducible characters by using their kernels. Also, we have presented the structure the solvable group G in which every nonlinear monolithic characters has same kernel

References

  • [1] Isaacs I. M., Character Theory of Finite Groups, Academic Press, New York, (1976.
  • [2] Seitz G.M., Finite groups having only one irreducible representation of degree greater than one. Proc. Am. Math. Soc., (19) (1968) 459-461.
  • [3] Manz O., Wolf T.R., Representations of Solvable Groups, London Mathematical Society Lecture Note Series, (185), Cambridge University Press, Cambridge (1993).
  • [4] Berkovich Y., Zhmud E. M., Characters of Finite Groups. Part 2, American Mathemetical Society, (1999).
  • [5] Berkovich Y., On Isaacs’ three character degrees theorem, Proc. Am. Math. Soc. 125 (3) (1997) 669-677.

Year 2022, Volume 43, Issue 3, 449 - 453, 30.09.2022
https://doi.org/10.17776/csj.904864

Abstract

References

  • [1] Isaacs I. M., Character Theory of Finite Groups, Academic Press, New York, (1976.
  • [2] Seitz G.M., Finite groups having only one irreducible representation of degree greater than one. Proc. Am. Math. Soc., (19) (1968) 459-461.
  • [3] Manz O., Wolf T.R., Representations of Solvable Groups, London Mathematical Society Lecture Note Series, (185), Cambridge University Press, Cambridge (1993).
  • [4] Berkovich Y., Zhmud E. M., Characters of Finite Groups. Part 2, American Mathemetical Society, (1999).
  • [5] Berkovich Y., On Isaacs’ three character degrees theorem, Proc. Am. Math. Soc. 125 (3) (1997) 669-677.

Details

Primary Language English
Subjects Mathematics
Journal Section Natural Sciences
Authors

Burcu ÇINARCI>
PİRİ REİS ÜNİVERSİTESİ
0000-0003-1202-0968
Türkiye


Temha ERKOÇ> (Primary Author)
İSTANBUL ÜNİVERSİTESİ
0000-0001-5437-3679
Türkiye

Supporting Institution TÜBİTAK
Project Number 119F295
Publication Date September 30, 2022
Application Date March 29, 2021
Acceptance Date May 6, 2022
Published in Issue Year 2022, Volume 43, Issue 3

Cite

APA Çınarcı, B. & Erkoç, T. (2022). A Study On the Kernels of Irreducible Characters of Finite Groups . Cumhuriyet Science Journal , 43 (3) , 449-453 . DOI: 10.17776/csj.904864