• Arhrafi, Ali-Reza (Department of Mathematics Faculty of Science University of Kashan)
  • Published : 2004.05.01


Let G be a finite group and N be a normal subgroup of G. We denote by ncc(N) the number of conjugacy classes of N in G and N is called n-decomposable, if ncc(N) = n. Set $K_{G}\;=\;\{ncc(N)$\mid$N{\lhd}G\}$. Let X be a non-empty subset of positive integers. A group G is called X-decomposable, if KG = X. In this paper we characterise the {1, 3, 4}-decomposable finite non-perfect groups. We prove that such a group is isomorphic to Small Group (36, 9), the $9^{th}$ group of order 36 in the small group library of GAP, a metabelian group of order $2^n{2{\frac{n-1}{2}}\;-\;1)$, in which n is odd positive integer and $2{\frac{n-1}{2}}\;-\;1$ is a Mersenne prime or a metabelian group of order $2^n(2{\frac{n}{3}}\;-\;1)$, where 3$\mid$n and $2\frac{n}{3}\;-\;1$ is a Mersenne prime. Moreover, we calculate the set $K_{G}$, for some finite group G.


  1. Vietnam J. Math. v.30 no.3 On Finite Groups Whose Every Normal Subgroup is a Union of the Same Number of Conjugacy Classes A.R.Ashrafi;H.Sahraei
  2. London Math. Soc. Lecture Note Ser. Subgroups which are a union of a given number of conjugacy classes A.R.Ashrafi;H.Sahraei
  3. Math. Slovaca v.53 no.4 On n-Decomposable Finite Groups A.R.Ashrafi;Y.Zhao
  4. On Finite Groups Whose Every Normal Subgroup is a Union of a Given Number of Conjugacy Classes A.R.Ashrafi
  5. Atlas of Finite Groups J.H.Conway;R.T.Curtis;S.P.Norton;R.A.Parker;R.A.Wilson
  6. Endliche Gruppen B.Huppert
  7. Comm. Algebra v.29 no.2 Subgroups which are the union of four conjugacy classes Udo Riese;M.A.Shahabi
  8. Grad. Texts in Math. (2nd ed.) v.80 A Course in the Theory of Groups Derek J. S. Robinson
  9. Grad. Texts in Math. v.148 An introduction to the theory of groups J.J.Rotman
  10. M.Sc. thesis, University of Kashan Subgroups which are a Union of Conjugacy Classes H.Sahraei
  11. Lehrstuhl De fur Mathematik GAP, Groups, Algorithms and Programming M.Schonert(et al.)
  12. Bull, Iranian Math. Soc. v.25 no.1 Subgroups which are the union of two conjugacy classes M.Shahryari;M.A.Shahabi
  13. J. Algebra v.207 Subgroups which are the union of three conjugate classes M.Shahryari;M.A.Shahabi
  14. J. Southwest Teachers College no.3 A class of special minimal normal subgroups(Chinese) Wujie Shi
  15. The Quantitative Structure of Groups and Related Topics Wujie Shi;Zhe-Xian Wan(ed.);Sheng-Ming Shi(ed.)
  16. Chinese Sci. Bull. v.37 A class of special finite groups Wujie Shi;C.Yang
  17. J. Southwest Teachers College v.9 no.1 A new characterization of $A_5$ and the finite groups in which every non-identity element has prime order(Chinese) Wujie Shi;Wenze Yang
  18. J. Chengdu University of Science and Technology no.4 A special class of normal subgroups(Chinese) Jing Wang

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