OD-CHARACTERIZATION OF ALMOST SIMPLE GROUPS RELATED TO U_6(2)

Let G be a finite group and π（G） = {pl,p2,…… ,pk} be the set of the primes dividing the order of G. We define its prime graph F（G） as follows. The vertex set of this graph is 7r（G）, and two distinct vertices p, q are joined by an edge if and only if pq ∈ πe（G）. In this case, we write p - q. For p ∈π（G）, put deg（p） ：= ｜{q ∈ π（G）｜p - q}｜, which is called the degree of p. We also define D（G） ：= （deg（p1）, deg（p2）,..., deg（pk））, where pl 〈 p2 〈 -……〈 pk, which is called the degree pattern of G. We say a group G is k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups with the same order and degree pattern as G. Specially, a l-fold OD-characterizable group is simply called an OD-characterizable group. Let L ：= U6（2）. In this article, we classify all finite groups with the same order and degree pattern as an almost simple groups related to L. In fact, we prove that L and L.2 are OD-characterizable, L.3 is 3-fold OD-characterizable, and L.S3 is 5-fold OD-characterizable.

A Kind of Diophantine Equations in Finite Simple Groups

In this paper, we prove that if p, q are distinct primes, (p,q)≡(1,7) (mod 12) and Legendres symbol pq=1 , then the equation 1+p a=2 bq c+2 dp eq f has only solutions of the form (a,b,c,d,e,f)=(t,0,0,0,t,0), where t is a non negative integer. We also give all solutions of a kind of generalized Ramanujan Nagell equations by using the theories of imaginary quadratic field and Pells equation.

Finite Groups Isospectral to Simple Groups

The spectrum of a finite group is the set of element orders of this group.The main goal of this paper is to survey results concerning recognition of finite simple groups by spectrum,in particular,to list all finite simple groups for which the recognition problem is solved.

OD-characterization of Almost Simple Groups Related to U_3(5)

Let G be a finite group with order ｜G｜=p1^α1p2^α2……pk^αk, where p1 〈 p2 〈……〈 Pk are prime numbers. One of the well-known simple graphs associated with G is the prime graph （or Gruenberg- Kegel graph） denoted .by г（G） （or GK（G））. This graph is constructed as follows： The vertex set of it is π（G） = {p1,p2,…,pk} and two vertices pi, pj with i≠j are adjacent by an edge （and we write pi - pj） if and only if G contains an element of order pipj. The degree deg（pi） of a vertex pj ∈π（G） is the number of edges incident on pi. We define D（G） ：= （deg（p1）, deg（p2）,..., deg（pk））, which is called the degree pattern of G. A group G is called k-fold OD-characterizable if there exist exactly k non- isomorphic groups H such that ｜H｜ = ｜G｜ and D（H） = D（G）. Moreover, a 1-fold OD-characterizable group is simply called OD-characterizable. Let L ：= U3（5） be the projective special unitary group. In this paper, we classify groups with the same order and degree pattern as an almost simple group related to L. In fact, we obtain that L and L.2 are OD-characterizable; L.3 is 3-fold OD-characterizable; L.S3 is 6-fold OD-characterizable.

Characterization of automorphism groups of sporadic simple groups

It is proved that all automorphism groups of the sporadic simple groups are characterized by their element orders and the group orders.

On the Characterizability of the Automorphism Groups of Sporadic Simple Groups by Their Element Orders

For G a finite group,π_e(G)denotes the set of orders of elements in G.If Ω is a subset of the set of natural numbers,h(Ω)stands for the number of isomorphism classes of finite groups with the same set Ω of element orders.We say that G is k-distinguishable if h(π_(G))=k<∞,otherwise G is called non-distinguishable.Usually,a 1-distinguishable group is called a characterizable group.It is shown that if M is a sporadic simple group different from M_(12),M_(22),J_2,He,Suz,M^cL and O'N, then Aut(M)is charaeterizable by its dement orders.It is also proved that if M is isomorphic to M_(12),M_(22),He,Suz or O'N,then h(π_e(Aut(M)))∈{1,∞}.

SU(3) simple group model and single top production at the e^-γ colliders

In the framework of the SU（3） simple group model, we consider the single top quark production process e^-γ →vebt^-. We find that the correction effects on the process mainly come from the terms of the tree-level Wqq＇ couplings. In the reasonable parameter space of the SU（3） simple group model, the deviation of the total production cross section σ^tot from its SM value is larger than 5%, which might be detected in the future high energy linear e^＋e^- collider （LC） experiments.

The Characterization of Finite Simple Groups,L_3(3^(2m-1))(m 2),by Their Element Orders

In this paper the following theorem is proved： Every group L3（q） for q = 3^（2m-1）（m≥2） is characterized by its set of element orders.

Coleman Automorphisms of Extensions of Finite Characteristically Simple Groups by Some Finite Groups

Let G be an extension of a finite characteristically simple group by an abelian group or a finite simple group.It is shown that every Coleman automorphism of G is an inner automorphism.Interest in such automorphisms arises from the study of the normalizer problem for integral group rings.

Recognizing by Spectrum for the Automorphism Groups of Sporadic Simple Groups

The spectrum of a finite group is the set of its element orders,and two groups are said to be isospectral if they have the same spectra.A finite group G is said to be recognizable by spectrum,if every finite group isospectral with G is isomorphic to G.We prove that if S is one of the sporadic simple groups M^(c)L,M_(12),M_(22),He,Suz and O'N,then Aut(S)is recognizable by spectrum.This finishes the proof of the recognizability by spectrum of the automorphism groups of all sporadic simple groups,except J_(2).

Characterization of Finite Simple Groups With Abelian Sylow 2-Subgroups

For the non-Abelian simple groups with Abelian Sylow 2-subgroups J. H. Walterhas proved the following famous theorem. Lemma 1. If F is a non-Ablian simple group with Abelian Sylow 2-subgroups, thenone of the following holds:（i）F≌PSL（2,Q）,q】3,q≡3,5（mod 8） or q=2n,n≥2;（ii）F≌J1;（iii）F≌R（q）,q=32m+1,m≥1.Let G be a finite group and let πe（G） denote the set of all orders of elements

Finite Group with Two Supersolvable Subgroups of Coprime Indices

In this paper, we obtain some classification theorems of finite simple groups with two subgroups of coprime indices which are both supersolvable or one supersolvable and the other nilpotent. Using these classification theorems, we prove some sufficient conditions of finite solvable groups. Finally, we provide a supplement of Doerks Theorem.

The Characterization of A5

The characterization of A 5 is obtained through the method of calculation.The main result is described as the following: 1)The order of A 5 is one,two,three or five. 2)The element of A 5 is divided into five conjugate classes. 3)There are fifty and nine subgroup in A 5 and we can obtain one produce element in every subgroup. 4)There are nine conjugate classes in the subgroup of A 5 .

Simple Kac-Moody groups with trivial Schur multipliers

We obtain new examples of simple Kac-Moody groups with trivial Schur multipliers.

A Characterization of Almost Simple K_3-groups

It is a well-known fact that characters of a finite group can give important information about the group＇s structure. Also it was proved by the third author of this article that a finite simple group can be uniquely determined by its character table. Here the authors attempt to investigate how to characterize a finite almost simple group by using less information of its character table, and successfully characterize the almost simple K3-groups by their orders and at most three irreducible character degrees of their character tables.

Basic 3-Transposittions of Unitary Group Un(2)

We aim to study maximal pairwise commuting sets of 3-transpositions(transvections)of the simple unitary group U_(n)(2)over GF(4),and to construct designs from these sets.Any maximal set of pairwise commuting 3-transpositions is called a basic set of transpositions.Let G=U_(n)(2).It is well known that G is a 3-transposition group with the set D,the conjugacy class consisting of its transvections,as the set of 3-transpositions.Let L be a set of basic transpositions in D.We give general descriptions of L and 1-(ν,κ,λ)designs D=(P,B),with P=D and B={L^(9)|g∈G}.The parameters k=|L|,λ and further properties of D are determined.We also,as examples,apply the method to the unitary simple groups U_(4)(2),U_(5)(2),U_(6)(2),U_(7)(2),U_(8)(2)and U_(9)(2).

A Characterization of L_n(q) by the Normalizers' Orders of Their Sylow Subgroups

Let G be a finite group. If |NG（R1）|=|NLn（q）（R2）| for every prime r, where R1∈Sylr G and R2∈Sylr（Ln（q））, then G≌Ln（q）.

Thompson＇s conjecture for alternating group of degree 22

For a finite group G, it is denoted by N（G） the set of conjugacy class sizes of G. In 1980s, J. G. Thompson posed the following conjecture： if L is a finite nonabelian simple group, G is a finite group with trivial center, and N（G） = N（L）, then L and G are isomorphic. In this paper, it is proved that Thompson＇s conjecture is true for the alternating group A22 with connected prime graph.

Properties of A Class of Groups on Zm

In this paper,we obtain the factorization of direct production and order of group GL（n,Z_m） in a simple method.Then we generalize some properties of GL（2,Z_p） proposed by Huppert,and prove that the group GL（2,Z_z^y） is solvable.We also prove that group GL（n,Z_p）is solvable if and only if GL（n,Z_p） is solvable,and list the generators of groups GL（n,Z_p） and SL（n,Z_p）.At last,we prove that PSL（2,Z_p）（ p〉3） and PSL（n,Z_p） （ n〉3） are simple.

The Normalizer Property for Integral Group Rings of Holomorphs of Finite Groups with Trivial Center

Let G=Hol(H)be the holomorph of a finite group H.If there is a prime q dividing|H|such that every q-central automorphism of H is inner and Z(H)=1,then every Coleman automorphism of G is inner.In particular,the normalizer property holds for G.