In this paper, the automorphism group of a generalized extraspecial p-group G is determined, where p is a prime number. Assume that |G| = p 2n+m and |ζG| = p m , where n 1 and m 2. (1) When p is odd, let Aut G G = {...In this paper, the automorphism group of a generalized extraspecial p-group G is determined, where p is a prime number. Assume that |G| = p 2n+m and |ζG| = p m , where n 1 and m 2. (1) When p is odd, let Aut G G = {α∈ AutG | α acts trivially on G }. Then Aut G G⊿AutG and AutG/Aut G G≌Z p-1 . Furthermore, (i) If G is of exponent p m , then Aut G G/InnG≌Sp(2n, p) × Z p m-1 . (ii) If G is of exponent p m+1 , then Aut G G/InnG≌ (K Sp(2n-2, p))×Z p m-1 , where K is an extraspecial p-group of order p 2n-1 . In particular, Aut G G/InnG≌ Z p × Z p m-1 when n = 1. (2) When p = 2, then, (i) If G is of exponent 2 m , then AutG≌ Sp(2n, 2) × Z 2 × Z 2 m-2 . In particular, when n = 1, |AutG| = 3 · 2 m+2 . None of the Sylow subgroups of AutG is normal, and each of the Sylow 2-subgroups of AutG is isomorphic to H K, where H = Z 2 × Z 2 × Z 2 × Z 2 m-2 , K = Z 2 . (ii) If G is of exponent 2 m+1 , then AutG≌ (I Sp(2n-2, 2)) × Z 2 × Z 2 m-2 , where I is an elementary abelian 2-group of order 2 2n-1 . In particular, when n = 1, |AutG| = 2 m+2 and AutG≌ H K, where H = Z 2 × Z 2 × Z 2 m-1 , K = Z 2 .展开更多
All the symplectic matrices possessing a fixed eigenvalue ω on the unit circle form a hypersurface in the real symplectic group Sp(2n). This paper is devoted to the study of the topological structures of this hypersu...All the symplectic matrices possessing a fixed eigenvalue ω on the unit circle form a hypersurface in the real symplectic group Sp(2n). This paper is devoted to the study of the topological structures of this hypersurface and its complement in Sp(2n).展开更多
A finite p-group P is called resistant if, for any finite group G having P as a Sylow p-group,the normalizer N_G(P) controls p-fusion in G. Let P be a central extension as 1→ Z_(p^m)→ P→ Z_p × · · &#...A finite p-group P is called resistant if, for any finite group G having P as a Sylow p-group,the normalizer N_G(P) controls p-fusion in G. Let P be a central extension as 1→ Z_(p^m)→ P→ Z_p × · · · × Z_p→1,and |P'|≤p,m≥2. The purpose of this paper is to prove that P is resistant.展开更多
Let ASG（2v ＋ l, v; Fq） be the （2v ＋l）-dimensional affine-singular symplectic space over the finite field Fq and ASp2v＋l,v（Fq） be the affine-singular symplectic group of degree 2v ＋ l over Fq. Let O be any or...Let ASG（2v ＋ l, v; Fq） be the （2v ＋l）-dimensional affine-singular symplectic space over the finite field Fq and ASp2v＋l,v（Fq） be the affine-singular symplectic group of degree 2v ＋ l over Fq. Let O be any orbit of flats under ASp2v＋l,v（Fq）. Denote by J the set of all flats which are joins of flats in O such that O LJ and assume the join of the empty set of flats in ASG（2v ＋ l, v;Fq） is φ. Ordering LJ by ordinary or reverse inclusion, then two lattices axe obtained. This paper firstly studies the inclusion relations between different lattices, then determines a characterization of flats contained in a given lattice LJ, when the lattices form geometric lattice, lastly gives the characteristic polynomial of LJ.展开更多
The main purpose of this paper is to provide a systematic geometric frame for generalized controlled Hamiltonian systems. The pseudo-Poisson manifold and the ω-manifold are proposed as the statespace of the generaliz...The main purpose of this paper is to provide a systematic geometric frame for generalized controlled Hamiltonian systems. The pseudo-Poisson manifold and the ω-manifold are proposed as the statespace of the generalized controlled Hamiltonian systems. A Lie group, calledN-group, and its Lie algebra, calledN-algebra, are introduced for the structure analysis of the systems. Some properties, including spectrum, structure-preservation, etc. are investigated. As an example the theoretical results are applied to power systems. The stabilization of excitation systems is investigated.展开更多
In this paper, we consider the relation of the Morse index of a closedgeodesic with the Maslov-type index of a path in a symplectic group. More precisely, for a closedgeodesic c on a Riemannian manifold M with its lin...In this paper, we consider the relation of the Morse index of a closedgeodesic with the Maslov-type index of a path in a symplectic group. More precisely, for a closedgeodesic c on a Riemannian manifold M with its linear Poincare map P (a symplectic matrix), weconstruct a symplectic path γ(t) starting from identity I and ending at P, such that the Morseindex of the closed geodesic c equals the Maslov-type index of γ. As an application of this result,we study the parity of the Morse index of any closed geodesic.展开更多
基金supported by National Natural Science Foundation of China (Grant No.10671058)Doctor Foundation of Henan University of Technology (Grant No. 2009BS029)
文摘In this paper, the automorphism group of a generalized extraspecial p-group G is determined, where p is a prime number. Assume that |G| = p 2n+m and |ζG| = p m , where n 1 and m 2. (1) When p is odd, let Aut G G = {α∈ AutG | α acts trivially on G }. Then Aut G G⊿AutG and AutG/Aut G G≌Z p-1 . Furthermore, (i) If G is of exponent p m , then Aut G G/InnG≌Sp(2n, p) × Z p m-1 . (ii) If G is of exponent p m+1 , then Aut G G/InnG≌ (K Sp(2n-2, p))×Z p m-1 , where K is an extraspecial p-group of order p 2n-1 . In particular, Aut G G/InnG≌ Z p × Z p m-1 when n = 1. (2) When p = 2, then, (i) If G is of exponent 2 m , then AutG≌ Sp(2n, 2) × Z 2 × Z 2 m-2 . In particular, when n = 1, |AutG| = 3 · 2 m+2 . None of the Sylow subgroups of AutG is normal, and each of the Sylow 2-subgroups of AutG is isomorphic to H K, where H = Z 2 × Z 2 × Z 2 × Z 2 m-2 , K = Z 2 . (ii) If G is of exponent 2 m+1 , then AutG≌ (I Sp(2n-2, 2)) × Z 2 × Z 2 m-2 , where I is an elementary abelian 2-group of order 2 2n-1 . In particular, when n = 1, |AutG| = 2 m+2 and AutG≌ H K, where H = Z 2 × Z 2 × Z 2 m-1 , K = Z 2 .
基金Partially supported by NNSFMCSEC of ChinaQiu Shi Sci Tech. Foundation
文摘All the symplectic matrices possessing a fixed eigenvalue ω on the unit circle form a hypersurface in the real symplectic group Sp(2n). This paper is devoted to the study of the topological structures of this hypersurface and its complement in Sp(2n).
基金Supported by NSFC(Grant Nos.11371154,11301150 and 11601121)Natural Science Foundation of Henan Province of China(Grant Nos.142300410134,162300410066)
文摘A finite p-group P is called resistant if, for any finite group G having P as a Sylow p-group,the normalizer N_G(P) controls p-fusion in G. Let P be a central extension as 1→ Z_(p^m)→ P→ Z_p × · · · × Z_p→1,and |P'|≤p,m≥2. The purpose of this paper is to prove that P is resistant.
基金Supported by the National Natural Science Foundation of China under Grant No.61179026 and No.11701558
文摘Let ASG（2v ＋ l, v; Fq） be the （2v ＋l）-dimensional affine-singular symplectic space over the finite field Fq and ASp2v＋l,v（Fq） be the affine-singular symplectic group of degree 2v ＋ l over Fq. Let O be any orbit of flats under ASp2v＋l,v（Fq）. Denote by J the set of all flats which are joins of flats in O such that O LJ and assume the join of the empty set of flats in ASG（2v ＋ l, v;Fq） is φ. Ordering LJ by ordinary or reverse inclusion, then two lattices axe obtained. This paper firstly studies the inclusion relations between different lattices, then determines a characterization of flats contained in a given lattice LJ, when the lattices form geometric lattice, lastly gives the characteristic polynomial of LJ.
文摘The main purpose of this paper is to provide a systematic geometric frame for generalized controlled Hamiltonian systems. The pseudo-Poisson manifold and the ω-manifold are proposed as the statespace of the generalized controlled Hamiltonian systems. A Lie group, calledN-group, and its Lie algebra, calledN-algebra, are introduced for the structure analysis of the systems. Some properties, including spectrum, structure-preservation, etc. are investigated. As an example the theoretical results are applied to power systems. The stabilization of excitation systems is investigated.
基金Project 10071040 supported by NNSF,200014 supported by Excellent.Ph.D.Funds of ME of ChinaPMC Key Lab.of ME of China
文摘In this paper, we consider the relation of the Morse index of a closedgeodesic with the Maslov-type index of a path in a symplectic group. More precisely, for a closedgeodesic c on a Riemannian manifold M with its linear Poincare map P (a symplectic matrix), weconstruct a symplectic path γ(t) starting from identity I and ending at P, such that the Morseindex of the closed geodesic c equals the Maslov-type index of γ. As an application of this result,we study the parity of the Morse index of any closed geodesic.
