scholarly journals On normal graph of a finite group

Filomat ◽  
2018 ◽  
Vol 32 (11) ◽  
pp. 4047-4059
Author(s):  
Ali Ashrafi ◽  
Fatemeh Koorepazan-Moftakhar
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Group Structure ◽  
Simple Graph ◽  
Conjugacy Classes ◽  
Open Questions ◽  
A Finite Group ◽  
Proper Normal Subgroup ◽  
Identical Character ◽  
Normal Graph

Suppose G is a finite group and C(G) denotes the set of all conjugacy classes of G. The normal graph of G, N(G), is a finite simple graph such that V(N(G)) = C(G). Two conjugacy classes A and B in C(G) are adjacent if and only if there is a proper normal subgroup N such that A U B ? N. The aim of this paper is to study the normal graph of a finite group G. It is proved, among other things, that the groups with identical character table have isomorphic normal graphs and so this new graph associated to a group has good relationship by its group structure. The normal graphs of some classes of finite groups are also obtained and some open questions are posed.

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 On the minimum degree of the power graph of a finite cyclic group

2020 ◽  
pp. 2150044
Author(s):  
Ramesh Prasad Panda ◽  
Kamal Lochan Patra ◽  
Binod Kumar Sahoo
Keyword(s):  
Finite Group ◽  
Cyclic Group ◽  
Finite Groups ◽  
Minimum Degree ◽  
Simple Graph ◽  
The Other ◽  
Prime Divisors ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

The power graph [Formula: see text] of a finite group [Formula: see text] is the undirected simple graph whose vertex set is [Formula: see text], in which two distinct vertices are adjacent if one of them is an integral power of the other. For an integer [Formula: see text], let [Formula: see text] denote the cyclic group of order [Formula: see text] and let [Formula: see text] be the number of distinct prime divisors of [Formula: see text]. The minimum degree [Formula: see text] of [Formula: see text] is known for [Formula: see text], see [R. P. Panda and K. V. Krishna, On the minimum degree, edge-connectivity and connectivity of power graphs of finite groups, Comm. Algebra 46(7) (2018) 3182–3197]. For [Formula: see text], under certain conditions involving the prime divisors of [Formula: see text], we identify at most [Formula: see text] vertices such that [Formula: see text] is equal to the degree of at least one of these vertices. If [Formula: see text], or that [Formula: see text] is a product of distinct primes, we are able to identify two such vertices without any condition on the prime divisors of [Formula: see text].

ON MINIMAL WEAKLY s-SUPPLEMENTED SUBGROUPS OF FINITE GROUPS

2011 ◽  
Vol 10 (05) ◽  
pp. 811-820 ◽  
Author(s):  
YANGMING LI ◽  
BAOJUN LI
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Positive Answer ◽  
Minimal Subgroup ◽  
Open Questions ◽  
Permutable Subgroups ◽  
A Finite Group

Suppose G is a finite group and H is subgroup of G. H is said to be s-permutable in G if HGp = GpH for any Sylow p-subgroup Gp of G; H is called weakly s-supplemented subgroup of G if there is a subgroup T of G such that G = HT and H ∩ T ≤ HsG, where HsG is the subgroup of H generated by all those subgroups of H which are s-permutable in G. We investigate the influence of minimal weakly s-supplemented subgroups on the structure of finite groups and generalize some recent results. Furthermore, we give a positive answer in the minimal subgroup case for Skiba's Open Questions in [On weakly s-permutable subgroups of finite groups, J. Algebra315 (2007) 192–209].

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.

Some sufficient conditions on solvability of finite groups

2016 ◽  
Vol 15 (03) ◽  
pp. 1650057 ◽  
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 ◽  
A Finite Group

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

2012 ◽  
Vol 153 (2) ◽  
pp. 281-318 ◽  
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.

Generalized Frattini Subgroups of Finite Groups. II

1969 ◽  
Vol 21 ◽  
pp. 418-429 ◽  
Author(s):  
James C. Beidleman
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Proper Subgroup ◽  
Subnormal Subgroup ◽  
Normal Subgroups ◽  
Frattini Subgroup ◽  
A Finite Group ◽  
Proper Normal Subgroup ◽  
Equivalent Conditions

The theory of generalized Frattini subgroups of a finite group is continued in this paper. Several equivalent conditions are given for a proper normal subgroup H of a finite group G to be a generalized Frattini subgroup of G. One such condition on H is that K is nilpotent for each normal subgroup K of G such that K/H is nilpotent. From this result, it follows that the weakly hyper-central normal subgroups of a finite non-nilpotent group G are generalized Frattini subgroups of G.Let H be a generalized Frattini subgroup of G and let K be a subnormal subgroup of G which properly contains H. Then H is a generalized Frattini subgroup of K.Let ϕ(G) be the Frattini subgroup of G. Suppose that G/ϕ(G) is nonnilpotent, but every proper subgroup of G/ϕ(G) is nilpotent. Then ϕ(G) is the unique maximal generalized Frattini subgroup of G.

Finite groups with few non-normal subgroups

2015 ◽  
Vol 14 (04) ◽  
pp. 1550057
Author(s):  
Jiakuan Lu ◽  
Linna Pang ◽  
Yanyan Qiu
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Conjugacy Classes ◽  
Normal Subgroups ◽  
A Finite Group

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)|.

Sign in / Sign up

Export Citation Format

Share Document