Siberian Mathematical Journal

, Volume 53, Issue 3, pp 444–449 | Cite as

On finite x-decomposable groups for X = {1, 2, 4}

Article

Abstract

A normal subgroup N of a finite group G is called an n-decomposable subgroup if N is a union of n distinct conjugacy classes of G. Each finite nonabelian nonperfect group is proved to be isomorphic to Q12, or Z2 × A4, or G = 〈a, b, c | a11 = b5 = c2 = 1, b−1ab = a4, c−1ac = a−1, c−1bc = b−1〉 if every nontrivial normal subgroup is 2- or 4-decomposable.

Keywords

n-decomposable X-decomposable G-conjugacy class 

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References

  1. 1.
    López A. V. and López J. V., “Classification of finite groups according to the number of conjugacy classes. II,” Israel J. Math., 56, 188–221 (1986).MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    López A. V. and de Elguea L. O., “On the number of conjugacy classes in a finite group,” J. Algebra, 115, 46–74 (1988).MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Chillag D. and Herzog M., “On the length of the conjugacy classes of finite groups,” J. Algebra, 131, No. 1, 110–125 (1990).MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Ashrafi A. R. and Sahraei H., “Subgroups which are a union of a given number of conjugacy classes,” in: Groups St. Andrews 2001 in Oxford, V. 1. (Oxford, 2001), Cambridge Univ. Press, Cambridge, 2001, pp. 101–109 (London Math. Soc. Lect. Note Ser.; V. 304).Google Scholar
  5. 5.
    Ashrafi A. R. and Sahraei H., “On finite groups whose every normal subgroup is a union of the same number of conjugacy classes,” Vietnam J. Math., 30, No. 3, 289–294 (2002).MathSciNetMATHGoogle Scholar
  6. 6.
    Ashrafi A. R. and Zhao Y., “On 5- and 6-decomposable finite groups,” Math. Slovaca, 53, No. 4, 373–383 (2003).MathSciNetMATHGoogle Scholar
  7. 7.
    Shahryari M. and Shahabi M. A., “Subgroups which are the union of two conjugacy classes,” Bull. Iranian Math. Soc., 25, No. 1, 59–71 (1999).MathSciNetMATHGoogle Scholar
  8. 8.
    Shahryari M. and Shahabi M. A., “Subgroups which are the union of three conjugacy classes,” J. Algebra, 207, 326–332 (1998).MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Riese U. and Shahabi M. A., “Subgroups which are the union of four conjugacy classes,” Comm. Algebra, 29, No. 2, 695–701 (2001).MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Shi W. J., “A class of special minimal normal subgroups,” J. Southwest Teachers College, 9, 9–13 (1984).Google Scholar
  11. 11.
    Wang J., “A special class of normal subgroups,” J. Chengdu Univ. Sci. Tech., 4, 115–119 (1987).Google Scholar
  12. 12.
    Ashrafi A. R. and Shi W. J., “On 7- and 8-decomposable finite groups,” Math. Slovaca, 55, No. 3, 253–262 (2005).MathSciNetMATHGoogle Scholar
  13. 13.
    Ashrafi A. R. and Shi W. J., “On 9- and 10-decomposable finite groups,” J. Appl. Math. Comput., 26, 169–182 (2008).MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Ashrafi A. R., “On decomposability of finite groups,” J. Korean Math. Sci., 41, No. 3, 479–487 (2004).MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Ashrafi A. R. and Venkataraman G., “On finite groups whose every proper normal subgroup is a union of a given number of conjugacy classes,” Indian Acad. Sci. (Math. Sci.), 114, No. 3, 217–224 (2004).MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Kurzweil H. and Stellmacher B., The Theory of Finite Groups: An Introduction, Springer-Verlag, New York, Berlin, and Heidelberg (2004).MATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  1. 1.Shanghai UniversityShanghaiP. R. China
  2. 2.Institute of MathematicsYunnnan UniversityKunmingP. R. China

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