scholarly journals Prime powers as conjugacy class lengths of π-elements

2004 ◽  
Vol 69 (2) ◽  
pp. 317-325 ◽  
Author(s):  
Antonio Beltrán ◽  
María José Felipe
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Normal Structure ◽  
Conjugacy Classes ◽  
First Case ◽  
A Finite Group

Let G be a finite group and π an arbitrary set of primes. We investigate the structure of G when the lengths of the conjugacy classes of its π-elements are prime powers. Under this condition, we show that such lengths are either powers of just one prime or exactly {1,qa, rb}, with q and r two distinct primes lying in π and a, b > 0. In the first case, we obtain certain properties of the normal structure of G, and in the second one, we provide a characterisation of the structure of G.

ON THE REGULAR GRAPH RELATED TO THE G-CONJUGACY CLASSES

2021 ◽  
pp. 1-5
Author(s):  
SH. RAHIMI ◽  
Z. AKHLAGHI
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Regular Graph ◽  
Simple Graph ◽  
Conjugacy Classes ◽  
A Finite Group

Abstract Given a finite group G with a normal subgroup N, the simple graph $\Gamma _{\textit {G}}( \textit {N} )$ is a graph whose vertices are of the form $|x^G|$ , where $x\in {N\setminus {Z(G)}}$ and $x^G$ is the G-conjugacy class of N containing the element x. Two vertices $|x^G|$ and $|y^G|$ are adjacent if they are not coprime. We prove that, if $\Gamma _G(N)$ is a connected incomplete regular graph, then $N= P \times {A}$ where P is a p-group, for some prime p, $A\leq {Z(G)}$ and $\textbf {Z}(N)\not = N\cap \textbf {Z}(G)$ .

scholarly journals The Lengths of Certain Real Conjugacy Classes and the Related Prime Graph

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2060
Author(s):  
Siqiang Yang ◽  
Xianhua Li
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Prime Graph ◽  
Conjugacy Classes ◽  
New Type ◽  
A Finite Group

Let G be a finite group. In this paper, we study how certain arithmetical conditions on the conjugacy class lengths of real elements of G influence the structure of G. In particular, a new type of prime graph is introduced and studied. We obtain a series of theorems which generalize some existed results.

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

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.

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 NORMAL SUBGROUPS WHOSE CONJUGACY CLASS GRAPH HAS DIAMETER THREE

2016 ◽  
Vol 94 (2) ◽  
pp. 266-272
Author(s):  
ANTONIO BELTRÁN ◽  
MARÍA JOSÉ FELIPE ◽  
CARMEN MELCHOR
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Conjugacy Classes ◽  
Normal Subgroups ◽  
A Finite Group ◽  
Class Graph

Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. We determine the structure of $N$ when the diameter of the graph associated to the $G$-conjugacy classes contained in $N$ is as large as possible, that is, equal to three.

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.

Conjugacy classes of double covers of monomial groups

1984 ◽  
Vol 96 (2) ◽  
pp. 195-201 ◽  
Author(s):  
John F. Humphreys
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Symmetric Group ◽  
Double Cover ◽  
Conjugacy Classes ◽  
Alternating Group ◽  
Double Covers ◽  
Monomial Group ◽  
The Subject ◽  
A Finite Group

Let G be a finite group, Sn be the symmetric group on n symbols and An be the corresponding alternating group. The conjugacy classes of the wreath product GSn (or monomial group as it is sometimes known) and the conjugacy classes of GAn have been described by Kerber (see [2] and [3]). The group Sn has a double cover n so that the faithful complex representations of this double cover may be regarded as protective representations of Sn. In Section 2, a particular double cover for GSn is constructed, the faithful complex representations of this group being the subject of a joint article with Peter Hoffman[1]. In the present paper, our task is to determine whether a conjugacy class of GSn corresponds to one or to two conjugacy classes in the double cover of GSn (and similarly for GAn). The main results, Theorems 1 and 2, are stated precisely in Section 2 and proved in Sections 3 and 4 respectively. The case when G = 1 provides classical results of Schur ([5], Satz IV). When G is a cyclic group, Read [4] has determined the conjugacy classes, not just for our particular double cover, but for all possible double covers of GSn.

scholarly journals Sign Conjugacy Classes of the Symmetric Groups

10.37236/5060 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Lucia Morotti
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Irreducible Character ◽  
Symmetric Groups ◽  
Conjugacy Classes ◽  
Class C ◽  
A Finite Group

A conjugacy class $C$ of a finite group $G$ is a sign conjugacy class if every irreducible character of $G$ takes value 0, 1 or -1 on $C$. In this paper we classify the sign conjugacy classes of the symmetric groups and thereby verify a conjecture of Olsson.

scholarly journals On the Number of Conjugacy Classes of a Finite Group

1993 ◽  
Vol 160 (2) ◽  
pp. 441-460 ◽  
Author(s):  
L.G. Kovacs ◽  
G.R. Robinson
Keyword(s):  
Finite Group ◽  
Conjugacy Classes ◽  
A Finite Group
Sign in / Sign up

Export Citation Format

Share Document