The Related Properties of Generalized Orthogonal Group in Specific Normed Linear Spaces

Firstly, in the general normed linear space, the concepts of generalized isosceles orthogonal group, generalized Birkhoff orthogonal group, generalized Roberts orthogonal group, strong Birkhoff orthogonal group and generalized orthogonal basis are introduced. Secondly, the conclusion that any two nonzero generalized orthogonal groups must be linearly independent group is proven. And the existence of nonzero generalized orthogonal group and its linear correlation are discussed preliminarily, as well as some related properties of nonempty generalized orthogonal group in specific normed linear space namely the lp space.

COMPLETENESS OF THE PARABOLIC SUBGROUPS OF PROJECTIVE ORTHOGONAL GROUPS

All parabolic subgroups and Borel subgroups of PΩ（2m ＋ 1, F） over a linearable field F of characteristic 0 are shown to be complete groups, provided m 〉 3.

BERNSTEIN'S FIRST SUMMABLE OPERATORS ON THE SPECIAL REAL ORTHOGONAL GROUP SO(n)

We study the structure of Bernstein's first summable operators and show that they con- verge uniformly to continuons functions on the special real orthogonal group SO(n)in this paper.In addition,we have also discussed the approximation degree to a class of function Lipα(0<α≤1)on SO(n).

The Isomorphisms of Orthogonal Groups over Local Rings

1. Introduction Since O’Meara worked out the automorphisms of orthogonal groups Ωn(V) over fields in [4] by using well-known residual space method, many results about the isomorphisms and automorphisms of orthogonal groups over integral domains have been achieved. Refer to O’Meara, Hahn for example. B. R. McDonald, in [12], determined the automorphisms of O(V) over local rings with 2 a unit by using involutions.

The Essential 2-rank for Classical Groups

Let G be a symplectic or orthogonal group defined over a finite field with odd characteristic and let D≤G be a Sylow 2-subgroup.In this paper,we classify the essential 2-subgroups and determine the essential 2-rank of the Frobenius category FD(G).Together with the results of An–Dietrich and Cao–An–Zeng,this completes the work of essential subgroups and essential ranks of classical groups.

A Converse Theorem for Split SO_(2l) over Finite Fields

We prove a converse theorem for split even special orthogonal groups over finite fields.This is the only case left on converse theorems of classical groups and the difficulty is the existence of the outer automorphism. In this paper, we develop new ideas and overcome this difficulty.