conjugacy class size
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Author(s):  
M. Bianchi ◽  
E. Pacifici ◽  
R. D. Camina ◽  
Mark L. Lewis

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.





2020 ◽  
Vol 130 (1) ◽  
Author(s):  
Qinhui Jiang ◽  
Changguo Shao ◽  
Yan Zhao


2019 ◽  
Vol 22 (5) ◽  
pp. 927-932
Author(s):  
Shuqin Dong ◽  
Hongfei Pan ◽  
Long Miao

Abstract Let {\operatorname{acd}(G)} and {\operatorname{acs}(G)} denote the average irreducible character degree and the average conjugacy class size, respectively, of a finite group G. The object of this paper is to prove that if \operatorname{acd}(G)<2(p+1)/(p+3) , then G=O_{p}(G)\times O_{{p^{\prime}}}(G) , and that if \operatorname{acs}(G)<4p/(p\kern-1.0pt+\kern-1.0pt3) , then G=O_{p}(G)\kern-1.0pt\times\kern-1.0ptO_{{p^{\prime}}}(G) with {O_{p}(G)} abelian, where p is a prime.



2019 ◽  
Vol 22 (2) ◽  
pp. 267-276
Author(s):  
Guohua Qian ◽  
Yong Yang

AbstractLetHbe a nilpotent subgroup of a finite nonabelian groupG, let{\pi=\pi(|H|)}and let{{\operatorname{bcl}}(G)}be the largest conjugacy class size of the groupG. In the present paper, we show that{|HO_{\pi}(G)/O_{\pi}(G)|<{\operatorname{bcl}}(G)}.





2017 ◽  
Vol 9 (1) ◽  
pp. 5-12
Author(s):  
A. K. Asboei ◽  
R. Mohammadyari

Abstract In this paper, will show that a symmetric group of prime degree p ≥ 5 is recognizable by its order and a special conjugacy class size of (p − 1)!.



2015 ◽  
Vol 125 (4) ◽  
pp. 501-506
Author(s):  
YANHENG CHEN ◽  
GUIYUN CHEN


2013 ◽  
Vol 89 (3) ◽  
pp. 522-523 ◽  
Author(s):  
QINGJUN KONG ◽  
QINGFENG LIU


2013 ◽  
Vol 88 (2) ◽  
pp. 297-300 ◽  
Author(s):  
QINGJUN KONG ◽  
QINGFENG LIU

AbstractLet $G$ be a finite $p$-solvable group and let ${G}^{\ast } $ be the set of elements of primary and biprimary orders of $G$. Suppose that the conjugacy class sizes of ${G}^{\ast } $ are $\{ 1, {p}^{a} , n, {p}^{a} n\} $, where the prime $p$ divides the positive integer $n$ and ${p}^{a} $ does not divide $n$. Then $G$ is, up to central factors, a $\{ p, q\} $-group with $p$ and $q$ two distinct primes. In particular, $G$ is solvable.



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