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**Sample text**

The group H is abelian and isomorphic to the factor group of the direct product ˆ of m (= rank of Σ) copies of Fq× by a subgroup isomorphic to Z(G(q)). In addition, in general NG(q) (U ) = U H and NG(q) (V ) = V H, with U ∩ H = V ∩ H = 1. Likewise the Weyl group W of Σ is recoverable from the Steinberg relations. Again in the untwisted case if one sets N = nα (t) | α ∈ Σ, t ∈ Fq× , one has H N and N/H ∼ = W. , W is generated by m involutions wi , 1 ≤ i ≤ m, where m is the rank of Σ, and a presentation for W is aﬀorded by w1 , .

It applies to any linearly independent α, β ∈ Σ and to each xα ∈ Xα , xβ ∈ Xβ . 2) [xα , xβ ] = xγ . γ 11 The Lie rank is sometimes also called the twisted Lie rank. There is a second notion of Lie rank, sometimes called the untwisted Lie rank; the two notions coincide for the untwisted groups. The untwisted Lie rank of a twisted group G(q) is the subscript in the Lie notation for G(q), or equivalently the Lie rank of the ambient algebraic group; it is the Lie rank of the untwisted group which was twisted to form G(q).

3. Assume that X = Y A with Y normal in X and (|Y |, |A|) = 1. If A is abelian but not cyclic, then Y = CY (a) | a ∈ A# . 3 are applied with A a p-group. Finally, we consider the structure of p-constrained groups Y such that Op (Y ) = 1, that is, groups such that F ∗ (Y ) = Op (Y ). By a theorem of Borel and Tits [BuWi1], all p-locals in simple groups of Lie type deﬁned with respect to ﬁelds of characteristic p enjoy this property. Accordingly, when all p-locals in an arbitrary simple group X have this property, X is said to have characteristic p-type9 .