Finite groups with at most six vanishing conjugacy classes

2021 ◽  
pp. 2250076
Author(s):  
Sajjad M. Robati ◽  
M. R. Darafsheh
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Irreducible Character ◽  
Conjugacy Classes ◽  
Character Formula ◽  
Simple Groups ◽  
Almost Simple Groups ◽  
A Finite Group

Let [Formula: see text] be a finite group. We say that a conjugacy class of [Formula: see text] in [Formula: see text] is vanishing if there exists some irreducible character [Formula: see text] of [Formula: see text] such that [Formula: see text]. In this paper, we show that finite groups with at most six vanishing conjugacy classes are solvable or almost simple groups.

Weights for Covering Groups of Symmetric and Alternating Groups, р ≠ 2

1991 ◽  
Vol 43 (4) ◽  
pp. 792-813 ◽  
Author(s):  
G. O. Michler ◽  
J. B. Olsson
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Irreducible Character ◽  
Modular Representation ◽  
Modular Representation Theory ◽  
A Finite Group ◽  
Covering Groups ◽  
Fundamental Paper

In his fundamental paper [1] J. L. Alperin introduced the idea of a weight in modular representation theory of finite groups G. Let p be a prime. A p-subgroup R is called a radical subgroup of G if R = Op(NG(R)). An irreducible character φ of NG(R) is called a weight character if φ is trivial on R and belongs to a p-block of defect zero of NG(R)/R. The G-conjugacy class of the pair (R, φ) is a weight of G. Let b be the p-block of NG(R) containing φ, and let B be p-block of G. A weight (R, φ) is a B-weight for the block B of G if B = bG, which means that B and b correspond under the Brauer homomorphism. Alperin's conjecture on weights asserts that the number l*(B) of B-weights of a p-block B of a finite group G equals the number l(B) of modular characters of B.

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 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 𝑝-closedness and 𝑝-nilpotency of finite groups

2019 ◽  
Vol 22 (5) ◽  
pp. 927-932
Author(s):  
Shuqin Dong ◽  
Hongfei Pan ◽  
Long Miao
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Irreducible Character ◽  
Class Size ◽  
Character Degree ◽  
Conjugacy Class Size ◽  
A Finite Group ◽  
Irreducible Character Degree

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.

scholarly journals GENERATING MAXIMAL SUBGROUPS OF FINITE ALMOST SIMPLE GROUPS

2020 ◽  
Vol 8 ◽  
Author(s):  
ANDREA LUCCHINI ◽  
CLAUDE MARION ◽  
GARETH TRACEY
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Simple Group ◽  
Finite Groups ◽  
Upper Bounds ◽  
Maximal Subgroups ◽  
Simple Groups ◽  
Almost Simple Group ◽  
Almost Simple Groups ◽  
A Finite Group

For a finite group $G$ , let $d(G)$ denote the minimal number of elements required to generate $G$ . In this paper, we prove sharp upper bounds on $d(H)$ whenever $H$ is a maximal subgroup of a finite almost simple group. In particular, we show that $d(H)\leqslant 5$ and that $d(H)\geqslant 4$ if and only if $H$ occurs in a known list. This improves a result of Burness, Liebeck and Shalev. The method involves the theory of crowns in finite groups.

Constructing Representations of Finite Simple Groups and Covers

2006 ◽  
Vol 58 (1) ◽  
pp. 23-38 ◽  
Author(s):  
Vahid Dabbaghian-Abdoly
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Groups ◽  
Irreducible Character ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Simple Method ◽  
Covering Group ◽  
Representations Of Finite Groups ◽  
A Finite Group

AbstractLet G be a finite group and χ be an irreducible character of G. An efficient and simple method to construct representations of finite groups is applicable whenever G has a subgroup H such that χH has a linear constituent with multiplicity 1. In this paper we show (with a few exceptions) that if G is a simple group or a covering group of a simple group and χ is an irreducible character of G of degree less than 32, then there exists a subgroup H (often a Sylow subgroup) of G such that χH has a linear constituent with multiplicity 1.

The Largest Lengths of Conjugacy Classes of Finite Groups

2005 ◽  
Vol 12 (03) ◽  
pp. 531-534
Author(s):  
Liguo He ◽  
Guohua Qian
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Conjugacy Classes ◽  
A Finite Group

Let bcl (G) denote the largest conjugacy class length of a finite group G. In this note, we prove that if bcl (G)<p2 for a prime p, then |G:Op(G)|p≤p.

Groups whose vanishing elements are of odd order

2018 ◽  
Vol 17 (10) ◽  
pp. 1850186
Author(s):  
S. M. Robati
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Irreducible Character ◽  
Character Formula ◽  
Odd Order ◽  
Vanishing Elements ◽  
A Finite Group

Let [Formula: see text] be a finite group. We say that an element [Formula: see text] in [Formula: see text] is a vanishing element if there exists some irreducible character [Formula: see text] of [Formula: see text] such that [Formula: see text]. In this paper, we study the structure of finite groups whose vanishing elements are of odd order.

GROUPS WITH THE SAME CHARACTER DEGREES AS SPORADIC ALMOST SIMPLE GROUPS

2016 ◽  
Vol 94 (2) ◽  
pp. 254-265
Author(s):  
SEYED HASSAN ALAVI ◽  
ASHRAF DANESHKHAH ◽  
ALI JAFARI
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Irreducible Character ◽  
Abelian Subgroup ◽  
Character Degrees ◽  
Simple Groups ◽  
Sporadic Simple Group ◽  
Almost Simple Group ◽  
Almost Simple Groups ◽  
A Finite Group

Let$G$be a finite group and$\mathsf{cd}(G)$denote the set of complex irreducible character degrees of$G$. We prove that if$G$is a finite group and$H$is an almost simple group whose socle is a sporadic simple group$H_{0}$and such that$\mathsf{cd}(G)=\mathsf{cd}(H)$, then$G^{\prime }\cong H_{0}$and there exists an abelian subgroup$A$of$G$such that$G/A$is isomorphic to$H$. In view of Huppert’s conjecture, we also provide some examples to show that$G$is not necessarily a direct product of$A$and$H$, so that we cannot extend the conjecture to almost simple groups.

Sign in / Sign up

Export Citation Format

Share Document