【作者】 赵静；

【导师】 南基洙；

【作者基本信息】 大连理工大学 ， 基础数学， 2009， 博士

【摘要】 令G是群,子群B和N是G的BN-对,T=B∩N是N的正规子群,则商群W=N/T是一个有限反射群,设W的极小生成集为S,那么∑=∑（W,S）是一个Building.本文的目的是研究典型群中非经典BN-对的构造,以及BN-对的有理不变式域.另外,本文还研究了Z₂上不可约有限T-群的分类问题.第一章用矩阵的方法构造了有限域上典型群的非经典的BN-对,并给出了子群B和N的有理不变式域的生成元集.第二章找出了有限域上（广义）典型群的子群B,N以及T的有理不变式域的完整的代数无关的生成元集,并计算子了群B,N以及T的阶数.第三章讨论了Z₂上本质的不可约有限T-群的分类问题.我们利用T-群与Cartan矩阵的关系,在群同构意义下重新给出了群的生成元,并计算了群的阶数.第四章利用特征数为2的有限域上辛对合矩阵构造了一类Catersian认证码,计算了该码的所有参数.在假定信源和编码规则按照等概率均匀分布的条件下,给出了该认证码被成功模仿攻击的最大概率P_I和被成功替换攻击的最大概率P_S.更多还原

【Abstract】 Let G be a group, B and N be the BN-pairs of G, and T = B∩N is a normal subgroup of TV, then we get a finite reflection group W = N/T. Let S be a minimal generating set of W, thenΣ=Σ（W, S） is a Building. The purpose of this thesis is to study the non-classical construction of BN-pairs of classical groups, the rational invariants of subgroups B and N, and also we consider the classification of the finite irreducible T-groups over Z₂.In Chapter 1, we construct non-classical BN-pairs arising from the classical groups over finite fields. Furthermore, we give transcendence bases of the rational invariant field of subgroups B and N.In Chapter 2, we construct explicit transcendence bases of the rational invariant fields of the generalized classical groups and subgroups B, N and T, and we also compute the orders of them.In Chapter 3, we determine the structure of the finite effective irreducible T-groups over Z₂. Up to isomorphism, we give explicit generators of the T-groups by using the Cartan matrices, and we also compute the orders of them.In Chapter 4, we compute the number of the symplectic involutions over the finite field F with char F = 2, and also one Cartesian authentication code is obtained. Furthermore, its size parameters are computed completely. Assume that the coding rules are chosen according to a uniform probability, P_I and P_S denote the largest probabilities of a successful impersonation attack and a successful substitution attack respectively, then P_I and P_S are also computed.更多还原