On characterization of a finite group by the set of conjugacy class sizes

2021 ◽  
pp. 2250226
Author(s):  
Ilya Gorshkov
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
General Result ◽  
Conjugacy Class Sizes ◽  
Thompson’S Conjecture ◽  
A Finite Group

Let [Formula: see text] be a finite group and [Formula: see text] be the set of its conjugacy class sizes. In the 1980s, Thompson conjectured that the equality [Formula: see text], where [Formula: see text] and [Formula: see text] is simple, implies the isomorphism [Formula: see text]. In a series of papers of different authors, Thompson’s conjecture was proved. In this paper, we show that in some cases it is possible to omit the conditions [Formula: see text] and [Formula: see text] is simple and prove a more general result.

ON THOMPSON'S CONJECTURE FOR ALMOST SPORADIC SIMPLE GROUPS

2013 ◽  
Vol 13 (02) ◽  
pp. 1350089 ◽  
Author(s):  
LINGLI WANG ◽  
WUJIE SHI
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Conjugacy Class ◽  
Simple Group ◽  
Simple Groups ◽  
Conjugacy Class Sizes ◽  
Sporadic Simple Groups ◽  
Thompson’S Conjecture ◽  
A Finite Group

Let G be a non-abelian group and N(G) be the set of conjugacy class sizes of G. In 1980s, Thompson posed the following conjecture: if M is a finite non-abelian simple group, G is a finite group with Z(G) = 1 and N(G) = N(M), then M ≅ G. Here, we prove Thompson's conjecture holds for all almost sporadic simple groups.

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.

ON CONJUGACY CLASS SIZES AND CHARACTER DEGREES OF FINITE GROUPS

2013 ◽  
Vol 13 (02) ◽  
pp. 1350100 ◽  
Author(s):  
GUOHUA QIAN ◽  
YANMING WANG
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Nilpotent Group ◽  
Finite Groups ◽  
Prime Power ◽  
P Element ◽  
Character Degrees ◽  
Prime Power Order ◽  
Conjugacy Class Sizes ◽  
A Finite Group

Let p be a fixed prime, G a finite group and P a Sylow p-subgroup of G. The main results of this paper are as follows: (1) If gcd (p-1, |G|) = 1 and p2 does not divide |xG| for any p′-element x of prime power order, then G is a solvable p-nilpotent group and a Sylow p-subgroup of G/Op(G) is elementary abelian. (2) Suppose that G is p-solvable. If pp-1 does not divide |xG| for any element x of prime power order, then the p-length of G is at most one. (3) Suppose that G is p-solvable. If pp-1 does not divide χ(1) for any χ ∈ Irr (G), then both the p-length and p′-length of G are at most 2.

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).

scholarly journals Several Results on Conjugacy Class Sizes of Some Elements of Finite Groups

2021 ◽  
Vol 20 ◽  
pp. 361-367
Author(s):  
Yongcai Ren
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Conjugacy Class Sizes ◽  
A Finite Group

Let G be a finite group. For an element x of G, xG denotes the conjugacy class of x in G. |xG| is called the size of the conjugacy class xG. In this paper, we establish several results on conjugacy class sizes of some elements of finite groups. In addition, we give a simple and clearer proof of a known result.

The sub-class sizes of some elements being square free

2021 ◽  
Author(s):  
Xianhe Zhao ◽  
Yanyan Zhou ◽  
Ruifang Chen ◽  
Qin Huang
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Prime Power ◽  
Prime Factor ◽  
Subnormal Subgroup ◽  
Prime Power Order ◽  
Conjugacy Class Sizes ◽  
A Finite Group

Let [Formula: see text] be an element of a finite group [Formula: see text], and [Formula: see text] a prime factor of the order of [Formula: see text]. It is clear that there always exists a unique minimal subnormal subgroup containing [Formula: see text], say [Formula: see text]. We call the conjugacy class of [Formula: see text] in [Formula: see text] the sub-class of [Formula: see text] in [Formula: see text], see [G. Qian and Y. Yang, On sub-class sizes of finite groups, J. Aust. Math. Soc. (2020) 402–411]. In this paper, assume that [Formula: see text] is the product of the subgroups [Formula: see text] and [Formula: see text], we investigate the solvability, [Formula: see text]-nilpotence and supersolvability of the group [Formula: see text] under the condition that the sub-class sizes of prime power order elements in [Formula: see text] are [Formula: see text] free, [Formula: see text] free and square free, respectively, so that some known results relevant to conjugacy class sizes are generalized.

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.

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