CONJUGATE GRAPHS OF FINITE GROUPS

2012 ◽  
Vol 04 (02) ◽  
pp. 1250035 ◽  
Author(s):  
A. ERFANIAN ◽  
B. TOLUE
Keyword(s):  
Simple Group ◽  
Finite Groups ◽  
Nonabelian Group ◽  
Nonabelian Simple Group ◽  
Vertex Set ◽  
Thompson’S Conjecture ◽  
Central Factors ◽  
Nonabelian Groups

In this paper we introduce the conjugate graph [Formula: see text] associated to a nonabelian group G with vertex set G\Z(G) such that two distinct vertices join by an edge if they are conjugate. We show if [Formula: see text], where S is a finite nonabelian simple group which satisfy Thompson's conjecture, then G ≅ S. Further, if central factors of two nonabelian groups H and G are isomorphic and |Z(G)| = |Z(H)|, then H and G have isomorphic conjugate graphs.

A NEW CHARACTERIZATION OF U4(7) BY ITS NONCOMMUTING GRAPH

2009 ◽  
Vol 08 (01) ◽  
pp. 105-114 ◽  
Author(s):  
LIANGCAI ZHANG ◽  
WUJIE SHI
Keyword(s):  
Simple Group ◽  
Simple Groups ◽  
Nonabelian Simple Group ◽  
Prime Graphs ◽  
Vertex Set ◽  
Thompson’S Conjecture ◽  
A Finite Group ◽  
Noncommuting Graph ◽  
Almost All

Let G be a finite nonabelian group and associate a disoriented noncommuting graph ∇(G) with G as follows: the vertex set of ∇(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In 1987, J. G. Thompson gave the following conjecture.Thompson's Conjecture If G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying N(G) = N(M), then G ≅ M, where N(G) denotes the set of the sizes of the conjugacy classes of G.In 2006, A. Abdollahi, S. Akbari and H. R. Maimani put forward a conjecture in [1] as follows.AAM's Conjecture Let M be a finite nonabelian simple group and G a group such that ∇(G)≅ ∇ (M). Then G ≅ M.Even though both of the two conjectures are known to be true for all finite simple groups with nonconnected prime graphs, it is still unknown for almost all simple groups with connected prime graphs. In the present paper, we prove that the second conjecture is true for the projective special unitary simple group U4(7).

scholarly journals On the structure of finite groups isospectral to finite simple groups

2015 ◽  
Vol 18 (5) ◽  
Author(s):  
Mariya A. Grechkoseeva ◽  
Andrey V. Vasil'ev
Keyword(s):  
Simple Group ◽  
Finite Groups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Element Orders ◽  
Nonabelian Simple Group

AbstractFinite groups are said to be isospectral if they have the same sets of element orders. A finite nonabelian simple group

scholarly journals Some results on the main supergraph of finite groups

2020 ◽  
Vol 30 (2) ◽  
pp. 172-178
Author(s):  
A. K. Asboei ◽  
◽  
S. S. Salehi ◽  
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Groups ◽  
Sporadic Simple Group ◽  
Vertex Set ◽  
A Finite Group

Let G be a finite group. The main supergraph S(G) is a graph with vertex set G in which two vertices x and y are adjacent if and only if o(x)∣o(y) or o(y)∣o(x). In this paper, we will show that G≅PSL(2,p) or PGL(2,p) if and only if S(G)≅S(PSL(2,p)) or S(PGL(2,p)), respectively. Also, we will show that if M is a sporadic simple group, then G≅M if only if S(G)≅S(M).

scholarly journals On finite groups with the Cayley invariant property

1997 ◽  
Vol 56 (2) ◽  
pp. 253-261
Author(s):  
Cai Heng Li
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Groups ◽  
Cayley Graphs ◽  
Invariant Property ◽  
Frobenius Groups ◽  
Nonabelian Simple Group ◽  
A Finite Group ◽  
Ci Property

A finite group G is said to have the m-CI property if, for any two Cayley graphs Cay(G, S) and Cay(G, T) of valency m, Cay(G, S) ≅ Cay(G, T) implies Sσ = T for some automorphism σ of G. In this paper, we investigate finite groups with the m-CI property. We first construct groups with the 3-CI property but not with the 2-CI property, and then prove that a nonabelian simple group has the 3-CI property if and only if it is A5. Finally, for infinitely many values of m, we construct Frobenius groups with the m-CI property but not with the nontrivial k-CI property for any k < m.

scholarly journals On connectedness of power graphs of finite groups

2018 ◽  
Vol 17 (10) ◽  
pp. 1850184 ◽  
Author(s):  
Ramesh Prasad Panda ◽  
K. V. Krishna
Keyword(s):  
Finite Groups ◽  
Upper Bound ◽  
The Other ◽  
Number Of Components ◽  
Cyclic Groups ◽  
Power Graph ◽  
Vertex Set ◽  
Proper Power

The power graph of a group [Formula: see text] is the graph whose vertex set is [Formula: see text] and two distinct vertices are adjacent if one is a power of the other. This paper investigates the minimal separating sets of power graphs of finite groups. For power graphs of finite cyclic groups, certain minimal separating sets are obtained. Consequently, a sharp upper bound for their connectivity is supplied. Further, the components of proper power graphs of [Formula: see text]-groups are studied. In particular, the number of components of that of abelian [Formula: see text]-groups are determined.

Non-commuting graphs of certain almost simple groups

2019 ◽  
Vol 12 (05) ◽  
pp. 1950081
Author(s):  
M. Jahandideh ◽  
R. Modabernia ◽  
S. Shokrolahi
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Simple Group ◽  
Simple Groups ◽  
Almost Simple Group ◽  
Almost Simple Groups ◽  
Commuting Graph ◽  
Vertex Set

Let [Formula: see text] be a non-abelian finite group and [Formula: see text] be the center of [Formula: see text]. The non-commuting graph, [Formula: see text], associated to [Formula: see text] is the graph whose vertex set is [Formula: see text] and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text]. We conjecture that if [Formula: see text] is an almost simple group and [Formula: see text] is a non-abelian finite group such that [Formula: see text], then [Formula: see text]. Among other results, we prove that if [Formula: see text] is a certain almost simple group and [Formula: see text] is a non-abelian group with isomorphic non-commuting graphs, then [Formula: see text].

n-RECOGNITION BY PRIME GRAPH OF THE SIMPLE GROUP PSL(2,q)

2008 ◽  
Vol 07 (06) ◽  
pp. 735-748 ◽  
Author(s):  
BEHROOZ KHOSRAVI
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Prime Number ◽  
Finite Groups ◽  
Prime Graph ◽  
Split Extension ◽  
A Finite Group

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge if there is an element in G of order pq. It is proved that if p > 11 and p ≢ 1 (mod 12), then PSL(2,p) is uniquely determined by its prime graph. Also it is proved that if p > 7 is a prime number and Γ(G) = Γ(PSL(2,p2)), then G ≅ PSL(2,p2) or G ≅ PSL(2,p2).2, the non-split extension of PSL(2,p2) by ℤ2. In this paper as the main result we determine finite groups G such that Γ(G) = Γ(PSL(2,q)), where q = pk. As a consequence of our results we prove that if q = pk, k > 1 is odd and p is an odd prime number, then PSL(2,q) is uniquely determined by its prime graph and so these groups are characterizable by their prime graph.

On enhanced power graphs of finite groups

2018 ◽  
Vol 17 (08) ◽  
pp. 1850146 ◽  
Author(s):  
Sudip Bera ◽  
A. K. Bhuniya
Keyword(s):  
Finite Groups ◽  
Abelian Groups ◽  
Cyclic Subgroups ◽  
Power Graph ◽  
Vertex Set ◽  
One To One

Given a group [Formula: see text], the enhanced power graph of [Formula: see text], denoted by [Formula: see text], is the graph with vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are edge connected in [Formula: see text] if there exists [Formula: see text] such that [Formula: see text] and [Formula: see text] for some [Formula: see text]. Here, we show that the graph [Formula: see text] is complete if and only if [Formula: see text] is cyclic; and [Formula: see text] is Eulerian if and only if [Formula: see text] is odd. We characterize all abelian groups and all non-abelian [Formula: see text]-groups [Formula: see text] such that [Formula: see text] is dominatable. Besides, we show that there is a one-to-one correspondence between the maximal cliques in [Formula: see text] and the maximal cyclic subgroups of [Formula: see text].

scholarly journals Order divisor graphs of finite groups

2018 ◽  
Vol 26 (3) ◽  
pp. 29-40
Author(s):  
S. U. Rehman ◽  
A. Q. Baig ◽  
M. Imran ◽  
Z. U. Khan
Keyword(s):  
Graph Theory ◽  
Finite Groups ◽  
Algebraic Graph Theory ◽  
Vertex Set ◽  
Productive Area

AbstractThe interplay between groups and graphs have been the most famous and productive area of algebraic graph theory. In this paper, we introduce and study the graphs whose vertex set is group G such that two distinct vertices a and b having di erent orders are adjacent provided that o(a) divides o(b) or o(b) divides o(a).

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