scholarly journals On Thompson’s Conjecture for Alternating GroupsAp+3

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Shitian Liu ◽  
Yong Yang
Keyword(s):  
Finite Group ◽  
Prime Graph ◽  
Conjugacy Classes ◽  
Connected Components ◽  
Prime Divisors ◽  
Vertex Set ◽  
Thompson’S Conjecture ◽  
Trivial Center ◽  
A Finite Group

LetGbe a group. Denote byπ(G)the set of prime divisors of|G|. LetGK(G)be the graph with vertex setπ(G)such that two primespandqinπ(G)are joined by an edge ifGhas an element of orderp·q. We sets(G)to denote the number of connected components of the prime graphGK(G). Denote byN(G)the set of nonidentity orders of conjugacy classes of elements inG. Alavi and Daneshkhah proved that the groups,Anwheren=p,p+1,p+2withs(G)≥2, are characterized byN(G). As a development of these topics, we will prove that ifGis a finite group with trivial center andN(G)=N(Ap+3)withp+2composite, thenGis isomorphic toAp+3.

scholarly journals ON THE PRIME GRAPH OF SIMPLE GROUPS

2014 ◽  
Vol 91 (2) ◽  
pp. 227-240 ◽  
Author(s):  
TIMOTHY C. BURNESS ◽  
ELISA COVATO
Keyword(s):  
Finite Group ◽  
Proper Subgroup ◽  
Prime Graph ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Prime Divisors ◽  
Prime Graphs ◽  
Vertex Set ◽  
A Finite Group

AbstractLet $G$ be a finite group, let ${\it\pi}(G)$ be the set of prime divisors of $|G|$ and let ${\rm\Gamma}(G)$ be the prime graph of $G$. This graph has vertex set ${\it\pi}(G)$, and two vertices $r$ and $s$ are adjacent if and only if $G$ contains an element of order $rs$. Many properties of these graphs have been studied in recent years, with a particular focus on the prime graphs of finite simple groups. In this note, we determine the pairs $(G,H)$, where $G$ is simple and $H$ is a proper subgroup of $G$ such that ${\rm\Gamma}(G)={\rm\Gamma}(H)$.

Conditions on the Edges and Vertices of Non-commuting Graph

2015 ◽  
Vol 74 (1) ◽  
Author(s):  
M. Jahandideh ◽  
M. R. Darafsheh ◽  
N. H. Sarmin ◽  
S. M. S. Omer
Keyword(s):  
Finite Group ◽  
Conjugacy Classes ◽  
Commuting Graph ◽  
Vertex Set ◽  
A Finite Group

Abstract - Let G􀡳 be a non- abelian finite group. The non-commuting graph ,􀪡is defined as a graph with a vertex set􀡳 − G-Z(G)􀢆in which two vertices x􀢞 and y􀢟 are joined if and only if xy􀢞􀢟 ≠ yx􀢟􀢞.  In this paper, we invest some results on the number of edges set , the degree of avertex of non-commuting graph and the number of conjugacy classes of a finite group. In order that if 􀪡􀡳non-commuting graph of H ≅ non - commuting graph of G􀪡􀡴,H 􀡴 is afinite group, then |G􀡳| = |H􀡴| .

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].

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.

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.

scholarly journals Quasirecognition by Prime Graph of the Groups 2D2n(q) Where q < 105

2017 ◽  
Author(s):  
Hossein Moradi ◽  
Mohammad Reza Darafsheh ◽  
Ali Iranmanesh
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Prime Power ◽  
Prime Graph ◽  
Composition Factor ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Prime Divisors ◽  
A Finite Group

Let G be a finite group. The prime graph &Gamma;(G) of G is defined as follows: The set of vertices of&nbsp;&Gamma;(G) is the set of prime divisors of |G| and two distinct vertices p and p' are connected in &Gamma;(G), whenever G has an element of order pp'. A non-abelian simple group P is called recognizable by prime graph if for any finite group G with &Gamma;(G)=&Gamma;(P), G has a composition factor isomorphic to P. In&nbsp;[4] proved finite simple groups 2Dn(q), where&nbsp;n&nbsp;&ne; 4k are quasirecognizable by prime graph. Now in this paper we discuss the quasirecognizability by prime graph of the simple groups 2D2k(q), where&nbsp;k &ge; 9 and q is a prime power less than 105.

N-critical graph of finite groups

2021 ◽  
Author(s):  
Viachaslau I. Murashka
Keyword(s):  
Finite Group ◽  
Directed Graph ◽  
Finite Groups ◽  
Soluble Group ◽  
Prime Divisors ◽  
Nilpotent Length ◽  
Critical Graph ◽  
Vertex Set ◽  
A Finite Group

A Schmidt [Formula: see text]-group is a non-nilpotent [Formula: see text]-group whose proper subgroups are nilpotent and which has the normal Sylow [Formula: see text]-subgroup. The [Formula: see text]-critical graph [Formula: see text] of a finite group [Formula: see text] is a directed graph on the vertex set [Formula: see text] of all prime divisors of [Formula: see text] and [Formula: see text] is an edge of [Formula: see text] if and only if [Formula: see text] has a Schmidt [Formula: see text]-subgroup. The bounds of the nilpotent length of a soluble group are obtained in terms of its [Formula: see text]-critical graph. The structure of a soluble group with given [Formula: see text]-critical graph is obtained in terms of commutators. The connections between [Formula: see text]-critical and other graphs (Sylow, soluble, prime, commuting) of finite groups are found.

Lower bounds on conjugacy classes of non-nilpotent subgroups in a finite group

2014 ◽  
Vol 14 (03) ◽  
pp. 1550039 ◽  
Author(s):  
Wei Meng ◽  
Jiakuan Lu
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Lower Bounds ◽  
Conjugacy Classes ◽  
Finite Solvable Group ◽  
Prime Divisors ◽  
A Finite Group

For a finite group G, let γ(G) denote the number of conjugacy classes of all non-nilpotent subgroups of G, and let π(G) denote the set of the prime divisors of |G|. In this paper, we establish lower bounds on γ(G). In fact, we show that if G is a finite solvable group, then γ(G) = 0 or γ(G) ≥ 2|π(G)|-2, and if G is non-solvable, then γ(G) ≥ |π(G)| + 1. Both lower bounds are best possible.

RECOGNIZING SOME FINITE SIMPLE GROUPS BY NONCOMMUTING GRAPH

2012 ◽  
Vol 11 (04) ◽  
pp. 1250077 ◽  
Author(s):  
M. KHEIRABADI ◽  
A. R. MOGHADDAMFAR
Keyword(s):  
Finite Group ◽  
Prime Graph ◽  
Simple Groups ◽  
Linear Groups ◽  
Nonabelian Group ◽  
Projective Special Linear Groups ◽  
Prime Graphs ◽  
Vertex Set ◽  
Trivial Center ◽  
Noncommuting Graph

Let G be a nonabelian group. We define the noncommuting graph ∇(G) of G as follows: its vertex set is G\Z(G), the noncentral elements of G, and two distinct vertices x and y of ∇(G) are joined by an edge if and only if x and y do not commute as elements of G, i.e. [x, y] ≠ 1. The finite group L is said to be recognizable by noncommuting graph if, for every finite group G, ∇(G) ≅ ∇ (L) implies G ≅ L. In the present article, it is shown that the noncommuting graph of a group with trivial center can determine its prime graph. From this, the following theorem is derived. If two finite groups with trivial centers have isomorphic noncommuting graphs, then their prime graphs coincide. It is also proved that the projective special unitary groups U4(4), U4(8), U4(9), U4(11), U4(13), U4(16), U4(17) and the projective special linear groups L9(2), L16(2) are recognizable by noncommuting graph.

scholarly journals Commuting Involution Graphs for Certain Exceptional Groups of Lie Type

2021 ◽  
Author(s):  
Ali Aubad ◽  
Peter Rowley
Keyword(s):  
Finite Group ◽  
Conjugacy Classes ◽  
Exceptional Groups ◽  
Vertex Set ◽  
Characteristic Two ◽  
A Finite Group ◽  
Groups Of Lie Type

AbstractSuppose that G is a finite group and X is a G-conjugacy classes of involutions. The commuting involution graph $${\mathcal {C}}(G,X)$$ C ( G , X ) is the graph whose vertex set is X with $$x, y \in X$$ x , y ∈ X being joined if $$x \ne y$$ x ≠ y and $$xy = yx$$ x y = y x . Here for various exceptional Lie type groups of characteristic two we investigate their commuting involution graphs.

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