Conjugacy class sizes and solvability of finite groups

2013 ◽  
Vol 123 (2) ◽  
pp. 239-244 ◽  
Author(s):  
QINHUI JIANG ◽  
CHANGGUO SHAO
Keyword(s):  
Conjugacy Class ◽  
Finite Groups ◽  
Conjugacy Class Sizes

ON -PARTS OF CONJUGACY CLASS SIZES OF FINITE GROUPS

2018 ◽  
Vol 97 (3) ◽  
pp. 406-411 ◽  
Author(s):  
YONG YANG ◽  
GUOHUA QIAN
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Conjugacy Classes ◽  
Conjugacy Class Sizes ◽  
A Finite Group

Let $G$ be a finite group. Let $\operatorname{cl}(G)$ be the set of conjugacy classes of $G$ and let $\operatorname{ecl}_{p}(G)$ be the largest integer such that $p^{\operatorname{ecl}_{p}(G)}$ divides $|C|$ for some $C\in \operatorname{cl}(G)$. We prove the following results. If $\operatorname{ecl}_{p}(G)=1$, then $|G:F(G)|_{p}\leq p^{4}$ if $p\geq 3$. Moreover, if $G$ is solvable, then $|G:F(G)|_{p}\leq p^{2}$.

A note on 𝑝-parts of conjugacy class sizes of finite groups

2019 ◽  
Vol 22 (5) ◽  
pp. 933-940
Author(s):  
Jinbao Li ◽  
Yong Yang
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Conjugacy Classes ◽  
Conjugacy Class Sizes ◽  
A Finite Group

Abstract Let G be a finite group and p a prime. Let {\operatorname{cl}(G)} be the set of conjugacy classes of G, and let {\operatorname{ecl}_{p}(G)} be the largest integer such that {p^{\operatorname{ecl}_{p}(G)}} divides {|C|} for some {C\in\operatorname{cl}(G)} . We show that if {p\geq 3} and {\operatorname{ecl}_{p}(G)=1} , then {\lvert G\mskip 1.0mu \mathord{:}\mskip 1.0mu O_{p}(G)\rvert_{p}\leq p^{3}} . This improves the main result of Y. Yang and G. Qian, On p-parts of conjugacy class sizes of finite groups, Bull. Aust. Math. Soc. 97 2018, 3, 406–411.

scholarly journals On solvable groups with one vanishing class size

2020 ◽  
pp. 1-20
Author(s):  
M. Bianchi ◽  
E. Pacifici ◽  
R. D. Camina ◽  
Mark L. Lewis
Keyword(s):  
Conjugacy Class ◽  
Finite Groups ◽  
Class Size ◽  
Single Element ◽  
Conjugacy Class Sizes ◽  
Conjugacy Class Size ◽  
Odd Order ◽  
Vanishing Elements ◽  
A Finite Group

Let G be a finite group, and let cs(G) be the set of conjugacy class sizes of G. Recalling that an element g of G is called a vanishing element if there exists an irreducible character of G taking the value 0 on g, we consider one particular subset of cs(G), namely, the set vcs(G) whose elements are the conjugacy class sizes of the vanishing elements of G. Motivated by the results inBianchi et al. (2020, J. Group Theory, 23, 79–83), we describe the class of the finite groups G such that vcs(G) consists of a single element under the assumption that G is supersolvable or G has a normal Sylow 2-subgroup (in particular, groups of odd order are covered). As a particular case, we also get a characterization of finite groups having a single vanishing conjugacy class size which is either a prime power or square-free.

scholarly journals CONJUGACY CLASS SIZES OF CERTAIN DIRECT PRODUCTS

2012 ◽  
Vol 85 (2) ◽  
pp. 217-231
Author(s):  
CARLO CASOLO ◽  
ELISA MARIA TOMBARI
Keyword(s):  
Conjugacy Class ◽  
Finite Groups ◽  
Prime Divisor ◽  
Full Description ◽  
Direct Products ◽  
Conjugacy Class Sizes

AbstractWe consider finite groups in which, for all primes p, the p-part of the length of any conjugacy class is trivial or fixed. We obtain a full description in the case in which for each prime divisor p of the order of the group there exists a noncentral conjugacy class of p-power size.

THE INFLUENCE OF CONJUGACY CLASS SIZES ON THE STRUCTURE OF FINITE GROUPS: A SURVEY

2011 ◽  
Vol 04 (04) ◽  
pp. 559-588 ◽  
Author(s):  
A. R. Camina ◽  
R. D. Camina
Keyword(s):  
Conjugacy Class ◽  
Recent Work ◽  
Finite Groups ◽  
Conjugacy Classes ◽  
Conjugacy Class Sizes ◽  
Major Objective ◽  
Early Results ◽  
The Many

The importance of conjugacy classes for the structure of finite groups was recognised very early in the study of groups. In this survey we consider the results from the many articles which have developed this topic and examined the influence of conjugacy class sizes or the number of conjugacy classes on the structure of finite groups. Whilst we begin by mentioning the early results of Sylow and Burnside, our major objective is to highlight the more recent work and present some interesting questions which we hope will inspire further research.

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