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)=...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.展开更多
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...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=2<sup>n</sup>,n≥2;（ii）F≌J<sub>1</sub>;（iii）F≌R（q）,q=3<sup>2m+1</sup>,m≥1.Let G be a finite group and let π<sub>e</sub>（G） denote the set of all orders of elements展开更多
Let G be a finite group. If |N<sub>G</sub>（R<sub>1</sub>）|=|N<sub>L<sub>n</sub></sub>（q）（R<sub>2</sub>）| for every prime r, where R<sub>1</sub...Let G be a finite group. If |N<sub>G</sub>（R<sub>1</sub>）|=|N<sub>L<sub>n</sub></sub>（q）（R<sub>2</sub>）| for every prime r, where R<sub>1</sub>∈Syl<sub>r</sub> G and R<sub>2</sub>∈Syl<sub>r</sub>（L<sub>n</sub>（q））, then G≌L<sub>n</sub>（q）.展开更多
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_...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.展开更多
文摘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.
文摘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=2<sup>n</sup>,n≥2;（ii）F≌J<sub>1</sub>;（iii）F≌R（q）,q=3<sup>2m+1</sup>,m≥1.Let G be a finite group and let π<sub>e</sub>（G） denote the set of all orders of elements
文摘Let G be a finite group. If |N<sub>G</sub>（R<sub>1</sub>）|=|N<sub>L<sub>n</sub></sub>（q）（R<sub>2</sub>）| for every prime r, where R<sub>1</sub>∈Syl<sub>r</sub> G and R<sub>2</sub>∈Syl<sub>r</sub>（L<sub>n</sub>（q））, then G≌L<sub>n</sub>（q）.
文摘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.
