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.

Locally primitive Cayley graphs of finite simple groups

A graph Г is said to be G-locally primitive, where G is a subgroup of automorphisms of Г, if the stabiliser Ga of a vertex α acts primitively on the set Г( α ) of vertices of Г adjacent to α. For a finite non-abelian simple group L and a Cayley subset S of L, suppose that L ? G ? Aut( L), and the Cayley graph Г = Cay ( L, S) is G-locally primitive. In this paper we prove that L is a simple group of Lie type, and either the valency of Г is an add prine divisor of |Out(L)|, orL =PΩ 8 + (q) and Г has valency 4. In either cases, it is proved that the full automorphism group of Г is also almost simple with the same socle L.

A characterization of finite simple groups by the orders of solvable subgroups

Let G be a finite group and S be a finite simple group. In this paper, we prove that if G and S have the same sets of all orders of solvable subgroups, then G is isomorphic to S, or G and S are isomorphic to Bn(q), Cn(q), where n≥3 and q is odd. This gives a positive answer to the problem put forward by Abe and Iiyori.

The p-local ranks of finite simple groups with abelian Sylow p-subgroups

As a continuation of previous work, in this paper, we mainly study the finite simple groups which have abelian Sylow subgroups in term of p-local rank, especially a group theoretic characterization will be given.

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

Some Simple Groups Which are Determined by Their Orders and Degree-Patterns

Let G be a finite group and Irr(G)the set of all irreducible complex characters of G.Let cd(G)be the set of all irreducible complex character degrees of G and denote byρ(G)the set of all primes which divide some character degree of G.The character-prime graphΓ(G)associated to G is a simple undirected graph whose vertex set isρ(G)and there is an edge between two distinct primes p and q if and only if the product p q divides some character degree of G.We show that the finite nonabelian simple groups A_(7),J_(1),J_(3),J_(4),L_(3)(3)and U_(3)(4)are uniquely determined by their degree-patterns and orders.

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）.

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.

OD-Characterization of Alternating and Symmetric Groups of Degree p+5

In this paper,it is proved that all the alternating groups A_(p+5) are ODcharacterizable and the symmetric groups S_(p+5) are 3-fold OD-characterizable,where p + 4 is a composite number and p + 6 is a prime and 5≠p∈π(1000!).