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

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 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].

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).

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 $\textbf {Z}(N)\not = N\cap \textbf {Z}(G)$ .

Prime Divisors ofp-regular Conjugacy Class Sizes

2015 ◽  
Vol 43 (8) ◽  
pp. 3365-3371 ◽  
Author(s):  
Yang Liu ◽  
Ziqun Lu
Keyword(s):  
Conjugacy Class ◽  
Conjugacy Class Sizes ◽  
Prime Divisors

Numbers Of Conjugacy Class Sizes And Derived Lengths for A-Groups

1996 ◽  
Vol 39 (3) ◽  
pp. 346-351 ◽  
Author(s):  
Mary K. Marshall
Keyword(s):  
Conjugacy Class ◽  
Solvable Group ◽  
Conjugacy Classes ◽  
Finite Solvable Group ◽  
Conjugacy Class Sizes ◽  
Sylow Subgroups ◽  
Derived Length ◽  
Completely Reducible

AbstractAn A-group is a finite solvable group all of whose Sylow subgroups are abelian. In this paper, we are interested in bounding the derived length of an A-group G as a function of the number of distinct sizes of the conjugacy classes of G. Although we do not find a specific bound of this type, we do prove that such a bound exists. We also prove that if G is an A-group with a faithful and completely reducible G-module V, then the derived length of G is bounded by a function of the number of distinct orbit sizes under the action of G on V.

CONJUGACY CLASS SIZES FOR SOME 2-GROUPS OF NILPOTENCY CLASS TWO

2012 ◽  
Vol 57 (1) ◽  
Author(s):  
SHEILA ILANGOVAN ◽  
NOR HANIZA SARMIN
Keyword(s):  
Conjugacy Class ◽  
Nilpotency Class ◽  
Conjugacy Class Sizes

Dalam kertas ini, kita menyelidik ciri tak terturunkan dan panjang kelas konjugat bagi kumpulan–2 berpenjana–2 dengan kelas nilpoten 2. Panjang kelas konjugat bagi elemen x dalam kumpulan G adalah peringkat xG di mana xG ialah kelas konjugat yang mengandungi x. Kajian ini adalah berdasarkan pada klasifikasi kumpulan yang diberikan oleh Magidin pada tahun 2006. Kita akan membuktikan bahawa panjang kelas konjugat bagi G ialah 2ρ di mana 0 <= ρ <= γdan |G'| = 2γ.

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