Finite groups of conjugate rank 2

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 $|G:F(G)|_{p}\leq p^{2}$.

Finite groups with at most six vanishing conjugacy classes

10.1142/s0219498822500761 ◽

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 ◽

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.

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

Journal of Group Theory ◽

10.1515/jgth-2018-0211 ◽

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 ◽

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.

Some sufficient conditions on solvability of finite groups

10.1142/s0219498816500572 ◽

2016 ◽

Vol 15 (03) ◽

pp. 1650057 ◽

Cited By ~ 1

Author(s):

Wei Meng ◽

Jiakuan Lu ◽

Li Ma ◽

Wanqing Ma

Keyword(s):

Finite Group ◽

Finite Groups ◽

Sufficient Conditions ◽

Conjugacy Classes ◽

Maximal Subgroups ◽

Prime Divisors ◽

Abelian Subgroups ◽

For a finite group [Formula: see text], the symbol [Formula: see text] denotes the set of the prime divisors of [Formula: see text] denotes the number of conjugacy classes of maximal subgroups of [Formula: see text]. Let [Formula: see text] denote the number of conjugacy classes of non-abelian subgroups of [Formula: see text] and [Formula: see text] denote the number of conjugacy classes of all non-normal non-abelian subgroups of [Formula: see text]. In this paper, we consider the finite groups with [Formula: see text] or [Formula: see text]. We show these groups are solvable.

Weak Cayley tables and generalized centralizer rings of finite groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004112000205 ◽

2012 ◽

Vol 153 (2) ◽

pp. 281-318 ◽

Cited By ~ 7

Author(s):

STEPHEN P. HUMPHRIES ◽

EMMA L. RODE

Keyword(s):

Finite Group ◽

Finite Groups ◽

Conjugacy Classes ◽

Cayley Table ◽

A Finite Group ◽

Finite Conjugacy ◽

Relationship Of ◽

Cayley Tables ◽

The Relationship

AbstractFor a finite group G we study certain rings (k)G called k-S-rings, one for each k ≥ 1, where (1)G is the centraliser ring Z(ℂG) of G. These rings have the property that (k+1)G determines (k)G for all k ≥ 1. We study the relationship of (2)G with the weak Cayley table of G. We show that (2)G and the weak Cayley table together determine the sizes of the derived factors of G (noting that a result of Mattarei shows that (1)G = Z(ℂG) does not). We also show that (4)G determines G for any group G with finite conjugacy classes, thus giving an answer to a question of Brauer. We give a criteria for two groups to have the same 2-S-ring and a result guaranteeing that two groups have the same weak Cayley table. Using these results we find a pair of groups of order 512 that have the same weak Cayley table, are a Brauer pair, and have the same 2-S-ring.

Finite groups with few non-normal subgroups

10.1142/s0219498815500577 ◽

2015 ◽

Vol 14 (04) ◽

pp. 1550057

Author(s):

Jiakuan Lu ◽

Linna Pang ◽

Yanyan Qiu

Keyword(s):

Finite Group ◽

Finite Groups ◽

Conjugacy Classes ◽

Normal Subgroups ◽

For a finite group G, let v(G) denote the number of conjugacy classes of all non-normal subgroups of G, and let π(G) denote the set of primes dividing the order of G. In this note, we shall classify the finite groups G with v(G) ≤ |π(G)|.

An analogy between products of two conjugacy classes and products of two irreducible characters in finite groups

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500017922 ◽

1987 ◽

Vol 30 (1) ◽

pp. 7-22 ◽

Cited By ~ 9

Author(s):

Zvi Arad ◽

Elsa Fisman

Keyword(s):

Finite Group ◽

Finite Groups ◽

Conjugacy Classes ◽

Irreducible Characters ◽

Basic Concepts ◽

It is well-known that the number of irreducible characters of a finite group G is equal to the number of conjugate classes of G. The purpose of this article is to give some analogous properties between these basic concepts.

Finite groups with given quantitative surpersoluble subgroups

10.1142/s0219498820500772 ◽

2019 ◽

Vol 19 (04) ◽

pp. 2050077

Author(s):

Jiakuan Lu ◽

Jingjing Wang

Keyword(s):

Finite Group ◽

Finite Groups ◽

Conjugacy Classes ◽

The solubility of a finite group with less than [Formula: see text] supersoluble subgroups is confirmed in this paper. Moreover, we prove that a finite insoluble group has exactly [Formula: see text] supersoluble subgroups if and only if it is isomorphic to [Formula: see text]. Furthermore, it is shown that a finite group with less than [Formula: see text] conjugacy classes of supersoluble subgroups is soluble, and the only finite insoluble group with [Formula: see text] conjugacy classes of supersoluble subgroups is isomorphic to [Formula: see text].

Powers of irreducible characters and conjugacy classes in finite groups

10.1142/s0219498814500674 ◽

2014 ◽

Vol 13 (08) ◽

pp. 1450067 ◽

Cited By ~ 1

Author(s):

M. R. Darafsheh ◽

S. M. Robati

Keyword(s):

Finite Group ◽

Positive Integer ◽

Finite Groups ◽

Upper Bounds ◽

Conjugacy Classes ◽

Covering Number ◽

Irreducible Characters ◽

Let G be a finite group. We define the derived covering number and the derived character covering number of G, denoted respectively by dcn (G) and dccn (G), as the smallest positive integer n such that Cn = G′ for all non-central conjugacy classes C of G and Irr ((χn)G′) = Irr (G′) for all nonlinear irreducible characters χ of G, respectively. In this paper, we obtain some results on dcn and dccn for a finite group G, such as the existence of these numbers and upper bounds on them.