Advertisement

Characterization of \({\textit{PGL}}(2 , p^{2} ) \) by Order and Some Irreducible Character Degrees

  • Ali IranmaneshEmail author
  • Mozhgan Mokhtari
  • Abolfazl Tehranian
Original Paper
  • 7 Downloads

Abstract

In this paper, we determine all of finite groups whose order and the largest of their irreducible character degrees are the same as \({\textit{PGL}}( 2 , p^{2} ) \) for all odd prime numbers p. As a consequence, we show that the groups \({\textit{PGL}}( 2 , p^{2} ) \) are uniquely determined by the structure of their complex group algebras.

Keywords

Almost simple groups Character degree Order Projective general linear group Complex group algebras 

Mathematics Subject Classification

20C15 20D05 20D60 

Notes

Acknowledgements

The authors express their gratitude to the referees for very useful comments and suggestions. Partial support by the Center of Excellence of Algebraic Structures and its Applications of Tarbiat Modares University (CEAS) is gratefully acknowledged by the first author (AI)

References

  1. 1.
    Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Oxford University Press, Oxford (1985)zbMATHGoogle Scholar
  2. 2.
    Ebrahimzadeh, B., Iranmanesh, A., Mosaed, H.: A new characterization of ree group \( ^{2}G_{2}(q) \) by the order of group and the number of elements with the same order. Int. J. Group Theory 6(4), 1–6 (2017)MathSciNetGoogle Scholar
  3. 3.
    Heydari, S., Ahanjideh, N.: A characterization of \( PGL(2, p^{n}) \) by some irreducible complex character degrees. Publ. Inst. Math. 90(113), 257–264 (2016)CrossRefGoogle Scholar
  4. 4.
    Heydari, S., Ahanjideh, N.: Characterization of some simple \( K_{4} \)-groups by some irreducible complex character degrees. Int. J. Group Theory 5(2), 61–74 (2016)MathSciNetGoogle Scholar
  5. 5.
    Huppert, B.: Character Theory of Finite Groups. de Gruyter, Berlin (1998)CrossRefGoogle Scholar
  6. 6.
    Issac, I.M.: Character Theory of Finite Groups, Pure and Applied Mathemathics, vol. 69. Academic Press, New York (1976)Google Scholar
  7. 7.
    Jiang, Q., Shao, C.: Recognition of \( L_{2}(q) \) by its group order and largest irreducible character degree. Monatsh. Math. 176, 413–422 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Khosravi, B.: Groups with the same orders and large character degrees as \( PGL(2, 9) \). Quasigroups Relat. Syst. 21, 239–243 (2013)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Khosravi, B., Khosravi, B., Khosravi, B.: Recognition of \( PSL(2, p) \) by order and some information on its character degrees where \( p \) is a prime. Monatsh. Math. 175(2), 277–282 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Khosravi, B., Khosravi, B., Khosravi, B., Momen, Z.: A new characterization for the simple group \( PSL(2, p^{2}) \) by order and some character degrees. Czech. Math. J. 65(1), 271–280 (2015)CrossRefGoogle Scholar
  11. 11.
    Salehi Amiri, S.S., Khalili Asboei, A., Iranmanesh, A., Tehranian, A.: Quasirecogrition by prime graph of \( U_{3}(q) \) where \( 2< q=p^{\alpha }<100 \). Int. J. Group Theory 1(3), 51–66 (2012)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Shirjian, F., Iranmanesh, A.: Characterizing projective general unitary groups \( PGU_{3}(q^{2}) \) by their complex group algebras. Czech. Math. J. 67(142), 819–826 (2017)CrossRefGoogle Scholar
  13. 13.
    Shirjian, F., Iranmanesh, A.: Complex group algebras of almost simple groups with socle \( PSL_{n}(q) \). Commun. Algebra 46(2), 552–573 (2018)CrossRefGoogle Scholar
  14. 14.
    Tong-Viet, H.P.: Alternating and sporadic simple groups are determined by their character degrees. Algebras Represent. Theory 15, 379–389 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Tong-Viet, H.P.: Simple classical groups of Lie Type are determined by their character degrees. J. Algebra 357, 61–68 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Tong-Viet, H.P.: Simple exceptional groups of Lie Type are determined by their character degrees. Monatsh. Math. 166, 556–577 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Tong-Viet, H.P.: Symmetric groups are determined by their character degrees. J. Algebra 334, 275–284 (2011)MathSciNetCrossRefGoogle Scholar
  18. 18.
    White, D.: Character degrees of extensions of \( PSL(2, q) \) and \( SL(2, q) \). J. Group Theory 16, 1–33 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Xu, H., Yan, Y., Chen, G.: A new characterization of Mathieu groups by the order and one irreducible character degree. J. Inequal. Appl. 209, 1–6 (2013)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Xu, H., Chen, G., Yan, Y.: A new characterization of simple \( K_{3}\)-groups by their orders and large degrees of their irreducible characters. Commun. Algebra 42, 5374–5380 (2014)CrossRefGoogle Scholar
  21. 21.
    Yan, Y., Feng, Y., Li, L., Xu, H.: A new characterization of the alternating group \( A_{8} \) by its order and large degrees of its irreducible character. Ital. J. Pure Appl. Math. 36, 257–264 (2016)zbMATHGoogle Scholar

Copyright information

© Iranian Mathematical Society 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Mathematical SciencesTarbiat Modares UniversityTehranIran
  2. 2.Department of Mathematics, Science and Research BranchIslamic Azad UniversityTehranIran

Personalised recommendations