Buildings and Groups Ⅱ

This is the second part of a pedagogical introduction to the theory of buildings of Jacques Tits.We de ne a(B,N)pair and construct a building out of it.Then we give a description of Chevalley groups,their(B,N)pair and the associated buildings.We illustrates this theory with many examples from classical groups.

HOMOMORPHISMS BETWEEN CHEVALLEY GROUPS OF TYPES C_nAND G_2 OVER FINITE FIELDS

It is known that there exists an isogeny sort of Chevalley groups G (Σ, F) associated to any indecomposable root system Σ and any field F . In this paper the author determines all nontrivial homomorphisms from G(Σ, k) to G(Σ, K) when the root system Σ is of type C n or G 2 , and the fields k and K are finite fields of characteristic p .

Isomorphism Between Gal(E/F) and Gal(L(E)/L(F))

Let E be a field of finite extension of a perfect field F. We show that Gal(E/F) is isomorphic to Gal(L(E)/L(F)).