On Decomposability of Finite Groups
by
Ali Reza Ashrafi
Department of Mathematics, University of Kashan, Kashan, Iran

Let G be a finite group and A is a normal subgroup of G. We denote by ncc(A) the number of conjugacy classes of A in G and A is called n-decomposable, if ncc(A)=n. Set K_G = { ncc(A)  |  A \lhd G }. Let Y be a non-empty subset of positive integers. A group G is called Y-decomposable, if K_G = Y.

The author, in [1-6] characterized the Y-decomposable non-perfect finite groups, for Y = { 1, n }, n <= 10, Y = { 1, 2, 3 } and Y = { 1, 3, 4 }. In this talk, we report on this problem.
2000 Mathematics Subject Classification: 20E34, 20D10.
Keywords and phrases: Finite group, conjugacy class, X-decomposable group.

[1] A.R. Ashrafi and H. Sahraei, On Finite Groups Whose Every Normal Subgroup is a Union of the Same Number of Conjugacy Classes, Vietnam J. Math., 30:3(2002), 289-294.
[2] A.R. Ashrafi and Zhao Yaoqing, On 5- and 6-decomposable finite groups, Math. Slovaca, 53(4)(2003), 373-383.
[3] A.R. Ashrafi, and Wujie Shi, On 7- and 8-decomposable finite groups, to appear in Math Slovaca, 2004.
[4] A.R. Ashrafi, On Decomposability of finite groups, to appear in J. Korean Math Soc.
[5] A.R. Ashrafi, On Finite Groups Whose Every Normal Subgroup is a Union of a Given Number of Conjugacy Classes, to appear in Proc. Indian Acad. Sci. Math. Sci., 2004.
[6] A.R. Ashrafi, and Wujie Shi, On 9- and 10-decomposable finite groups, submitted.

Date received: April 22, 2004