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Tori in Quasi-Split Groups
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Let G be a connected quasi-split semisimple algebraic group over a field k. Let S be a maximal k-split torus and T=Z(S) its centraliser. Then T is a maximal k-torus in G. The normaliser N(S)of S is also the normaliser N(T) of T . All these groups are defined over k and hence also N(T)/Z(T)=W, a finite group. Suppose now that T'⊂G is any maximal k-torus. Then there is an element g∈G(ks)/ such that gTg-1=T' where ks is a separable closure of k. If G is the Galois group of ks over k and σ∈G, evidently σ(g)Tσ(g)-1=σ(gTg-1)=T'.
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