Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On finite groups whose every proper normal subgroup is a union of a
  given number of conjugacy classes   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 ${\cal K}_G = \{ncc(A)| A \lhd G \}$. Let
$X$ be a non-empty subset of positive integers. A group $G$ is called
$X$-decomposable, if ${\cal K}_G = X$.
  Ashrafi and his co-authors \cite{ash1,ash2,ash3,ash4,ash5} have characterized
the $X$-decomposable non-perfect finite groups for $X = \{1, n \}$ and $n \leq
10$. 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 $Z_6, D_8, Q_8, S_4$, SmallGroup(20, 3),
SmallGroup(24, 3), where SmallGroup$(m,n)$ denotes the $m$th group of order $n$
in the small group library of GAP \cite{gap}.

10
The categories of this paper are: q-bio
9
The categories of this paper are: nlin
11
The categories of this paper are: hep-lat
14
The categories of this paper are: math-ph
7
The categories of this paper are: hep-th
36
The categories of this paper are: cmp-lg
12
The categories of this paper are: nucl-th
1
The categories of this paper are: cs
20
The categories of this paper are: acc-phys
21
The categories of this paper are: adap-org
15
The categories of this paper are: nucl-ex
19
The categories of this paper are: eess
35
The categories of this paper are: mtrl-th
34
The categories of this paper are: chem-ph
33
The categories of this paper are: plasm-ph
4
The categories of this paper are: cond-mat