Result: On finite groups whose every proper normal subgroup is a union of a given number of conjugacy classes: On finite groups whose every proper normal subgroup is a union of a given number of conjugacy classes.

Title:
On finite groups whose every proper normal subgroup is a union of a given number of conjugacy classes: On finite groups whose every proper normal subgroup is a union of a given number of conjugacy classes.
Authors:
Ali Reza Ashrafi, Geetha Venkataraman
Source:
Proceedings Mathematical Sciences. 114:217-224
Publication Status:
Preprint
Publisher Information:
Springer Science and Business Media LLC, 2004.
Publication Year:
2004
Subject Terms:
numbers of conjugacy classes, unions of conjugacy classes, Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks, Group Theory (math.GR), 0102 computer and information sciences, 01 natural sciences, finite groups, normal subgroups, 20E34, 20D10, FOS: Mathematics, 0101 mathematics, Mathematics - Group Theory, Arithmetic and combinatorial problems involving abstract finite groups, Conjugacy classes for groups
Document Type:
Academic journal Article<br />Other literature type
File Description:
application/xml
Language:
English
ISSN:
0973-7685
0253-4142
DOI:
10.1007/bf02830000
DOI:
10.48550/arxiv.math/0503030
Access URL:
http://arxiv.org/pdf/math/0503030v1.pdf
http://arxiv.org/abs/math/0503030
https://zbmath.org/2109500
https://doi.org/10.1007/bf02830000
https://www.ias.ac.in/mathsci/vol114/aug2004/Pm2101.pdf
https://link.springer.com/article/10.1007%2FBF02830000
http://ui.adsabs.harvard.edu/abs/2005math......3030A/abstract
Rights:
Springer TDM
arXiv Non-Exclusive Distribution
Accession Number:
edsair.doi.dedup.....fc88369e5203c79e0b9d87c508a6fb8c
Database:
OpenAIRE

Further Information

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}.
8 pages