BN-pairs Of Classical Groups And The Classification Of The Finite T-Groups Over Finite Fields

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.