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).

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 Characterization of Finite Simple Groups by the Degrees of Vertices of Their Prime Graphs

2005 ◽  
Vol 12 (03) ◽  
pp. 431-442 ◽  
Author(s):  
A. R. Moghaddamfar ◽  
A. R. Zokayi ◽  
M. R. Darafsheh
Keyword(s):  
Finite Group ◽  
Prime Graph ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Natural Numbers ◽  
Prime Graphs ◽  
Sporadic Simple Groups ◽  
A Finite Group ◽  
Groups Of Lie Type

If G is a finite group, we define its prime graph Γ(G) 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, denoted by p~q, if there is an element in G of order pq. Assume [Formula: see text] with primes p1<p2<⋯<pkand natural numbers αi. For p∈π(G), let the degree of p be deg (p)=|{q∈π(G)|q~p}|, and D(G):=( deg (p1), deg (p2),…, deg (pk)). In this paper, we prove that if G is a finite group such that D(G)=D(M) and |G|=|M|, where M is one of the following simple groups: (1) sporadic simple groups, (2) alternating groups Apwith p and p-2 primes, (3) some simple groups of Lie type, then G≅M. Moreover, we show that if G is a finite group with OC (G)={29.39.5.7, 13}, then G≅S6(3) or O7(3), and finally, we show that if G is a finite group such that |G|=29.39.5.7.13 and D(G)=(3,2,2,1,0), then G≅S6(3) or O7(3).

RECOGNITION OF THE PROJECTIVE GENERAL LINEAR GROUP PGL(2, q) BY ITS NONCOMMUTING GRAPH

2011 ◽  
Vol 10 (02) ◽  
pp. 201-218 ◽  
Author(s):  
LIANGCAI ZHANG ◽  
WUJIE SHI
Keyword(s):  
Simple Group ◽  
Linear Group ◽  
General Linear Group ◽  
Simple Groups ◽  
General Linear ◽  
Almost Simple Group ◽  
Almost Simple Groups ◽  
Vertex Set ◽  
Noncommuting Graph ◽  
Projective General Linear Group

The noncommuting graph ∇(G) of a non-abelian group G is defined as follows. The vertex set of ∇(G) is G\Z(G) where Z(G) denotes the center of G and two vertices x and y are adjacent if and only if xy ≠ yx. It has been conjectured that if P is a finite non-abelian simple group and G is a group such that ∇(P) ≅ ∇(G), then G ≅ P. In the present paper, our aim is to consider this conjecture in the case of finite almost simple groups. In fact, we characterize the projective general linear group PGL (2, q) (q is a prime power), which is also an almost simple group, by its noncommuting graph.

Mersenne primes and solvable Sylow numbers

2016 ◽  
Vol 15 (09) ◽  
pp. 1650163
Author(s):  
Tian-Ze Li ◽  
Yan-Jun Liu
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Solvable Groups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Nonabelian Simple Group ◽  
Mersenne Primes ◽  
A Finite Group ◽  
Mersenne Prime

Let [Formula: see text] be a prime. The Sylow [Formula: see text]-number of a finite group [Formula: see text], which is the number of Sylow [Formula: see text]-subgroups of [Formula: see text], is called solvable if its [Formula: see text]-part is congruent to [Formula: see text] modulo [Formula: see text] for any prime [Formula: see text]. P. Hall showed that solvable groups only have solvable Sylow numbers, and M. Hall showed that the Sylow [Formula: see text]-number of a finite group is the product of two kinds of factors: of prime powers [Formula: see text] with [Formula: see text] (mod [Formula: see text]) and of the number of Sylow [Formula: see text]-subgroups in certain finite simple groups (involved in [Formula: see text]). These classical results lead to the investigation of solvable Sylow numbers of finite simple groups. In this paper, we show that a finite nonabelian simple group has only solvable Sylow numbers if and only if it is isomorphic to [Formula: see text] for [Formula: see text] a Mersenne prime.

ON THOMPSON'S CONJECTURE FOR ALMOST SPORADIC SIMPLE GROUPS

2013 ◽  
Vol 13 (02) ◽  
pp. 1350089 ◽  
Author(s):  
LINGLI WANG ◽  
WUJIE SHI
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Conjugacy Class ◽  
Simple Group ◽  
Simple Groups ◽  
Conjugacy Class Sizes ◽  
Sporadic Simple Groups ◽  
Thompson’S Conjecture ◽  
A Finite Group

Let G be a non-abelian group and N(G) be the set of conjugacy class sizes of G. In 1980s, Thompson posed the following conjecture: if M is a finite non-abelian simple group, G is a finite group with Z(G) = 1 and N(G) = N(M), then M ≅ G. Here, we prove Thompson's conjecture holds for all almost sporadic simple groups.

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 ON HUPPERT’S CONJECTURE FOR THE CONWAY AND FISCHER FAMILIES OF SPORADIC SIMPLE GROUPS

2013 ◽  
Vol 94 (3) ◽  
pp. 289-303 ◽  
Author(s):  
S. H. ALAVI ◽  
A. DANESHKHAH ◽  
H. P. TONG-VIET ◽  
T. P. WAKEFIELD
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Simple Group ◽  
Irreducible Character ◽  
Character Degrees ◽  
Simple Groups ◽  
Nonabelian Simple Group ◽  
Sporadic Simple Groups ◽  
A Finite Group

AbstractLet $G$ denote a finite group and $\mathrm{cd} (G)$ the set of irreducible character degrees of $G$. Huppert conjectured that if $H$ is a finite nonabelian simple group such that $\mathrm{cd} (G)= \mathrm{cd} (H)$, then $G\cong H\times A$, where $A$ is an abelian group. He verified the conjecture for many of the sporadic simple groups and we complete its verification for the remainder.

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 ${\rm\Gamma}(G)={\rm\Gamma}(H)$.

A CHARACTERIZATION OF SPORADIC SIMPLE GROUPS BY NSE AND ORDER

2012 ◽  
Vol 12 (02) ◽  
pp. 1250158 ◽  
Author(s):  
ALIREZA KHALILI ASBOEI ◽  
SEYED SADEGH SALEHI AMIRI ◽  
ALI IRANMANESH ◽  
ABOLFAZL TEHRANIAN
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Simple Groups ◽  
Sporadic Simple Group ◽  
Sporadic Simple Groups ◽  
A Finite Group

Let G be a finite group and nse (G) the set of numbers of elements with the same order in G. In this paper, we prove that if ∣G∣ = ∣S∣ and nse (G) = nse (S), where S is a sporadic simple group, then the finite group G is isomorphic to S.

scholarly journals Simple groups with the same prime graph as 2Dn(q)

2015 ◽  
Vol 98 (112) ◽  
pp. 251-263
Author(s):  
Behrooz Khosravi ◽  
A. Babai
Keyword(s):  
Positive Integer ◽  
Simple Group ◽  
Prime Graph ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Nonabelian Simple Group ◽  
Prime Graphs

In 2006, Vasil'ev posed the problem: Does there exist a positive integer k such that there are no k pairwise nonisomorphic nonabelian finite simple groups with the same graphs of primes? Conjecture: k = 5. In 2013, Zvezdina, confirmed the conjecture for the case when one of the groups is alternating. We continue this work and determine all nonabelian simple groups having the same prime graphs as the nonabelian simple group 2Dn(q).

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