On normal subgroups with consecutive G-class sizes

2016 ◽  
Vol 15 (08) ◽  
pp. 1650151
Author(s):  
Changguo Shao ◽  
Qinhui Jiang
Keyword(s):  
Group Theory ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Frobenius Group ◽  
Normal Subgroups ◽  
Conjugacy Class Sizes ◽  
Abelian Kernel ◽  
Consecutive Integers ◽  
Normal Formula

Let [Formula: see text] be a group and [Formula: see text] be a normal subgroup of [Formula: see text]. If the set [Formula: see text] is composed by consecutive integers, then [Formula: see text] is either nilpotent or a quasi-Frobenius group with abelian kernel and complements. This is a generalization of Theorem 2 of [A. Beltrán, M. J. Felipe and C. G. Shao, [Formula: see text]-divisibility of conjugacy class sizes and normal [Formula: see text]-complements, J. Group Theory 18 (2015) 133–141].

On the normal subgroup with coprime G-conjugacy class sizes

2011 ◽  
Vol 121 (4) ◽  
pp. 397-404 ◽  
Author(s):  
XIANHE ZHAO ◽  
GUIYUN CHEN ◽  
JIAOYUN SHI
Keyword(s):  
Normal Subgroup ◽  
Conjugacy Class ◽  
Conjugacy Class Sizes

On coprime G-conjugacy class sizes in a normal subgroup

2014 ◽  
Vol 30 (9) ◽  
pp. 1588-1594 ◽  
Author(s):  
Xian He Zhao ◽  
Hai Peng Qu ◽  
Gui Yun Chen
Keyword(s):  
Normal Subgroup ◽  
Conjugacy Class ◽  
Conjugacy Class Sizes

scholarly journals PGL2(q) cannot be determined by its cs

Filomat ◽  
2020 ◽  
Vol 34 (5) ◽  
pp. 1713-1719
Author(s):  
Neda Ahanjideh
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Prime Power ◽  
Simple Groups ◽  
Conjugacy Class Sizes ◽  
Direct Factor ◽  
Almost Simple Groups ◽  
Trivial Element ◽  
A Finite Group

For a finite group G, let Z(G) denote the center of G and cs*(G) be the set of non-trivial conjugacy class sizes of G. In this paper, we show that if G is a finite group such that for some odd prime power q ? 4, cs*(G) = cs*(PGL2(q)), then either G ? PGL2(q) X Z(G) or G contains a normal subgroup N and a non-trivial element t ? G such that N ? PSL2(q)X Z(G), t2 ? N and G = N. ?t?. This shows that the almost simple groups cannot be determined by their set of conjugacy class sizes (up to an abelian direct factor).

Simplicity of normal subgroups and conjugacy class sizes

2014 ◽  
Vol 175 (4) ◽  
pp. 485-490
Author(s):  
Antonio Beltrán ◽  
María José Felipe
Keyword(s):  
Conjugacy Class ◽  
Normal Subgroups ◽  
Conjugacy Class Sizes

COMPONENTS AND MINIMAL NORMAL SUBGROUPS OF FINITE AND PSEUDOFINITE GROUPS

2019 ◽  
Vol 84 (1) ◽  
pp. 290-300
Author(s):  
JOHN S. WILSON
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Minimal Normal Subgroup ◽  
Normal Subgroups ◽  
First Order ◽  
Order Language ◽  
Image Position ◽  
A Finite Group

AbstractIt is proved that there is a formula$\pi \left( {h,x} \right)$in the first-order language of group theory such that each component and each non-abelian minimal normal subgroup of a finite groupGis definable by$\pi \left( {h,x} \right)$for a suitable elementhofG; in other words, each such subgroup has the form$\left\{ {x|x\pi \left( {h,x} \right)} \right\}$for someh. A number of consequences for infinite models of the theory of finite groups are described.

scholarly journals NORMAL SUBGROUPS WHOSE CONJUGACY CLASS GRAPH HAS DIAMETER THREE

2016 ◽  
Vol 94 (2) ◽  
pp. 266-272
Author(s):  
ANTONIO BELTRÁN ◽  
MARÍA JOSÉ FELIPE ◽  
CARMEN MELCHOR
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Conjugacy Classes ◽  
Normal Subgroups ◽  
A Finite Group ◽  
Class Graph

Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. We determine the structure of $N$ when the diameter of the graph associated to the $G$-conjugacy classes contained in $N$ is as large as possible, that is, equal to three.

scholarly journals Finite groups with three conjugacy class sizes of certain elements

2015 ◽  
Vol 98 (112) ◽  
pp. 265-270
Author(s):  
Qinhui Jiang ◽  
Changguo Shao
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Conjugacy Class Sizes ◽  
Abelian Kernel ◽  
Coprime Integers ◽  
A Finite Group

Let G be a finite group. Let m and n be two positive coprime integers. We prove that the set of conjugacy class sizes of primary and biprimary elements of G is {1,m,n} if and only if G is quasi-Frobenius with abelian kernel and complement.

A note on class sizes of vanishing elements in finite groups

2021 ◽  
pp. 2250067
Author(s):  
Qingjun Kong ◽  
Shi Chen
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Prime Power ◽  
Prime Power Order ◽  
Normal Subgroups ◽  
Conjugacy Class Sizes ◽  
Vanishing Elements ◽  
A Finite Group

Let [Formula: see text] and [Formula: see text] be normal subgroups of a finite group [Formula: see text]. We obtain th supersolvability of a factorized group [Formula: see text], given that the conjugacy class sizes of vanishing elements of prime-power order in [Formula: see text] and [Formula: see text] are square-free.

ON THE REGULAR GRAPH RELATED TO THE G-CONJUGACY CLASSES

2021 ◽  
pp. 1-5
Author(s):  
SH. RAHIMI ◽  
Z. AKHLAGHI
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Regular Graph ◽  
Simple Graph ◽  
Conjugacy Classes ◽  
A Finite Group

Abstract Given a finite group G with a normal subgroup N, the simple graph $\Gamma _{\textit {G}}( \textit {N} )$ is a graph whose vertices are of the form $|x^G|$ , where $x\in {N\setminus {Z(G)}}$ and $x^G$ is the G-conjugacy class of N containing the element x. Two vertices $|x^G|$ and $|y^G|$ are adjacent if they are not coprime. We prove that, if $\Gamma _G(N)$ is a connected incomplete regular graph, then $N= P \times {A}$ where P is a p-group, for some prime p, $A\leq {Z(G)}$ and $\textbf {Z}(N)\not = N\cap \textbf {Z}(G)$ .

Sign in / Sign up

Export Citation Format

Share Document