On Finite Groups whose Every Proper Normal Subgroup is a Union of a Given Number of Conjugacy Classes

Proceedings Mathematical Sciences

Geetha Venkataraman²  Ali Reza Ashrafi¹ 

[1] Department of Mathematics, University of Kashan, Kashan, Iran$$;Department of Mathematics and Mathematical Sciences Foundation, St. Stephen’s College, Delhi 0 00, India$$

关键词: Finite group;    𝑛-decomposable subgroup;    conjugacy class;    𝑋-decomposable group.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

Let 𝐺 be a finite group and 𝐴 be a normal subgroup of 𝐺. We denote by $ncc(A)$ the number of 𝐺-conjugacy classes of 𝐴 and 𝐴 is called 𝑛-decomposable, if $ncc(A)=n$. Set $mathcal{K}_G={ncc(A)|Avartriangleleft G}$. Let 𝑋 be a non-empty subset of positive integers. A group 𝐺 is called 𝑋-decomposable, if $mathcal{K}_G=X$.Ashrafi and his co-authors [1–5] have characterized the 𝑋-decomposable non-perfect finite groups for $X={1,n}$ and 𝑛 ≤ 10. In this paper, we continue this problem and investigate the structure of 𝑋-decomposable non-perfect finite groups, for $X={1, 2, 3}$. We prove that such a group is isomorphic to $Z_6, D_8, Q_8, S_4$, Small Group (20,3), Small Group (24,3), where Small Group (𝑚, 𝑛) denotes the $m^{mathrm{th}}$ group of order 𝑛 in the small group library of GAP [11].

【 授权许可】

Unknown